WORKS OF 
PROFESSOR F. DeR. FURMAN 

PUBLISHED BY 

JOHN WILEY & SONS, Inc. 



VALVES AND VALVE GEARS 
Vol. I. Steam Engines and Steam Turbines. 

X + 253 pages, 6 by 9. Profusely illustrated with figures 
in the text. Cloth, $2.50 net. 
Vol. II. Gasoline, Gas and Oil Engines. 

. xi + 190 pages, 6 by 9, 170 figures. Cloth, $2.00 net. 



ELEMENTARY CAMS 

vi + 90 pages, 6 by 9. Illustrated. Cloth, $1.25 net. 



ELEMENTARY CAMS 



BY 

FRANKLIN DeRONDE :fURMAN, M.E. 

Professor of Mechanism and Machine Design 

at Stevens Institute of Technology 

Member of American Society of Mechanical Engineers 



FIRST EDITION 
FIRST THOUSAND 



NEW YORK 
JOHN WILEY & SONS, Inc. 

London : CHAPMAN & HALL, Limited 
1916 



TTzo6 



Copyright, 1916, by 
FRANKLIN DiRONDE FURMAN 



1,:'^ 



(•■; 



DEC 28 I9IG 



Publishers Printing Company 
207-217 West Twenty-fifth Street, New York 



'CI.A453313 A 



"i 



b\ 



''T^A 



PREFACE 

With the constant introduction of automatic machinery in 
practically the whole range of manufacturing industry, the matter of 
form and size of the cam calls strongly for a comprehensive method 
of technical analysis. Such analysis becomes more and more impor- 
tant as the automatic devices or machines are designed to operate 
at higher and higher speeds. 

Of the vast number of cams in use, in industrial manufacturing 
plants particularly, it is quite certain that a large majority have 
been formed entirely ''by eye" as a result of experience and suc- 
cessive trials and without recourse to technical analysis or computa- 
tion. The entire development of cam work has been practically 
individualistic without a common thread of principle running through 
it all. This is in sharp contrast to the technical development of 
the subject of toothed gearing, for example. Cams, considered on a 
scientific basis, have been more neglected in engineering books 
than any other of the important branches of mechanism. Even the 
nomenclature of the subject is in a chaotic state, and the same types 
of cams are known by various names in different localities — some of 
them not far apart. 

This book on Elementary Cams has been prepared with a view to 
gathering the various types of cams that are in common use in some 
comprehensive and orderly manner, and then pointing out how to 
design one of each class by selecting the form that will come nearest 
to giving the desired velocity and acceleration to the follower. 
Simple arithmetical computations are made in advance to determine 
the size of the cam necessary to avoid ''hard spots" when in service. 
The present work includes practically all of the base curves that are 
in common use in industrial manufacturing work, and will enable 
the cam designer to lay down his plans with a foreknowledge of what 
may be expected in each case. 

F. DeR. Furman. 

HoBOKEN, N. J., November, 1916. 



lU 



CONTENTS 



PAGES 

Section I. — Definitions and Classification 1-19 

Cams Follower Surfaces Radial or Disk Cams Side or 
Cylindrical Cams Conical and Spherical Cams 

Names of Cams — Periphery, Plate, Heart, Frog, Mushroom, Face, 
Wiper, RolUng, Yoke, Cylindrical, End, Double End, Box, Internal, 
Offset, Positive Drive, Single Acting, Double Acting, Step, Adjustable, 
Clamp, Strap, Dog, Carrier, Double Mounted, Multiple Mounted, 
Oscillating 

Definitions of Terms Used in the Solution of Cam Problems — Cam 
Chart, Cam Chart Diagram, Time Chart, Base Curve, Base Line, 
Pitch Line, Pitch Circle, Pitch Surface, Working Surface, Pitch Point, 
Pressure Angle 

Formula for Size of Cam for a Given Maximum Pressure Angle 
Table of Cam Factors for All Base Curves for Maximum Pressure 
Angles from 20° to 60° 

Section II. — Method of Construction of Base Curves in Common 

Use 20-24 

Straight Line Base Straight-Line Combination Curve Crank 
Curve Parabola EUiptical Curve 

Section III. — Cam Problems and Exercise Problems 25-74 

Problem 1, Empirical Design Problem 2, Technical Design. 
Advantages of Technical Design Problem 3, Single-Step Radial 
Cam, Pressure Angle Equal on Both Strokes Omission of Cam 
Chart Problem 4, Single-Step Radial Cam, Pressure Angles Unequal 
on Both Strokes 

Pressure Angle Increases as Pitch Size of Cam Decreases Change 
of Pressure Angles in Passing from Cam Chart to Cam Cam Con- 
sidered as Bent Chart. Base Line Angles Before and After Bending 

Limiting Size of Follower Roller Radius of Curvature of Non- 
Circular Arcs 

Problem 5, Double-Step Radial Cam Determination of Maximum 
Pressure Angle for a Multiple-Step Cam 

Problem 6, Cam with Offset Roller Follower Problem 7, Cam 
with Flat Surface Follower Limited Use of Cams with Flat Surface 
Followers 

Problem 8, Cam with Swinging Follower Arm, Roller Contact — 
Extremities of Swinging Arc on Radial Line Problem 9, Cam 
with Swinging Follower Arm, Roller Contact — Swinging Arc, Con- 
tinued, Passes Through Center of Cam Effect of Location of 
Swinging Follower Arm Relatively to the Cam 



VI CONTENTS 

PAGES 

Problem 10, Face Cam with Swinging Follower Problem 11, Cam 
with Swinging Follower Arm, Sliding Surface Contact Data Limited 
for Followers with SUding Surface Contact 

Problem 12, Toe and Wiper Cam Modifications of the Toe and 
Wiper Cam 

Problem 13, Single Disk Yoke Cam Limited AppHcation of 
Single Disk Yoke Cam Problem 14, Double Disk Yoke Cam 

Problem 15, Cyhndrical Cam with Follower that Moves in a Straight 
Line Refinements in Cyhndrical Cam Design Problem 16, 
Cyhndrical Cam with Swinging Follower Chart Method for Laying 
Out a Cyhndrical Cam with a Swinging Follower Arm 

Exercise Problems, la to 16a 

Section IV. — Timing and Interference of Cams 75-78 

Problem 17, Cam Timing and Interference Location of Key ways 
Exercise Problem 17a 

Section V. — Cams for Reproducing Given Curves or Figures . 79-87 
Problem 18, Cam Mechanism for Drawing an Elhpse Problem 

18a, Exercise Problem for Drawing Figure 8 Problem 19, Cam 

for Reproducing Handwriting Using Script Letters Ste 

Method of Subdividing Circles into Any Desired Number of Equal 

Parts 



ELEMENTARY CAMS 

SECTION L— DEFINITIONS AND CLASSIFICATION 



Definitions 



1. Cams are rotating or oscillating pieces of mechanism having 
specially formed surfaces against which a follower slides or rolls 
and thus receives a reciprocating or intermittent motion such as 
cannot be generally obtained by gear wheels or link motions. 

Various forms of cams are illustrated at C in Figs. 1 to 10. The 
follower in each case is shown at F, all having roller contact except 
the ones shown in Figs. 7 and 8. The former has a V edge and 
the latter a plane surface in contact with the cam and both have 
sliding action. 

2. Follower edges or rollers may have motion in a straight 
line as from D to G, Fig. 7, or in a curved path depending on suit- 
ably constructed guides or on swinging arms. The total range of 
travel of the follower may be accomplished by one continuous 
motion, or by several separate motions with intervals of rest. Each 
motion may be either constant or variable in velocity, and the time 
used by the motion may be greater or less, all according to the 
work the machine has to do and to the will of the designer. 

Classification 

3. Cams may be most simply, and at the same time most com- 
pletely, classified according to the motion of the follower with re- 
spect to the axis of the cam, as: 

(a) Radial or disk cams, in which the radial distance from the 
cam axis to the acting surface varies constantly during part or all of 
the cam cycle, according to the data. The follower edge or roller 
moves in all cases in a radial, or an approximately radial, direction 
with respect to the cam. Various forms of radial cams are illus- 
trated in Figs. 1, 2, 7, 8, and 9. 

(6) Side or cylindrical cams, in which the follower edge or 
roller moves parallel to the axis of the cam, or approximately in 

1 



ELEMENTARY CAMS 



this direction. Several types of side cams are shown in Figs. 3, 4, 
and 10. 

Nearly all the cams referred to in the above figures illustrating 
the two general classes of radial and side cams respectively have 
special or local trade names which will be pointed out in a succeed- 
ing paragraph. 

(c) Conical and (d) Spherical cams, in which the follower edge 
or roller moves in an inclined direction having both radial and 

longitudinal components 
with respect to the axis 
of the cam as illustrated 
in Figs. 5 and 6. 

4. Names or cams. 
Cams, in popular usage, 
have come to be known 
by a wide range of names, 
the same cam often being 
designated by a number 
of different names accord- 
ing to geographical loca- 
tion and personal prefer- 
ence and surroundings of 
the cam builder or user. 
This is an unfortunate con- 
dition, and in the general 
classification in the preced- 
ing paragraph an endeavor 
is made to establish a fun- 
damental basis for clarifying and simplifying the nomenclature 
of cams as much as possible. In a treatise of this kind, however, it 
is essential that, at least, the more common of the ordinary working 
terms be recognized and defined, and that the cams under their 
popular names be properly placed in the fundamental classification 
given in the preceding paragraph. 

The following specially named cams fall under the classifica- 
tion of radial cams: 

(e) Periphery cams, in which the acting surface is the periphery 
of the cam, as illustrated in Figs. 1, 7, and 9. While these are ex- 
amples of true periphery cams, it must be recorded that the cylin- 
drical grooved cam, shown in Fig. 3, is also known to some extent 
as a periphery cam, due no doubt to the fact that in designing this 














^ 




^ 




^ 




— 




irztrr 


- 




— 
































c 





End 



IEront 



Fig. 1. 



-Radial Cam and Follower, 
Roller Contact 



DEFINITIONS AND CLASSIFICATION 3 

cam the original layout for the contour of the groove is first made 
on a flat piece of paper, which is then wrapped on to the surface or 
''periphery" of the cylinder. Since the contour line of the groove 
which lies on the periphery is merely a guiding Hne for cutting the 
groove, and since the side surface of the groove is the working sur- 
face, it is, to say the least, a misnomer to designate such a cam as 
a periphery cam. 

(/) Plate cams, in which the working surface includes the full 
360°, and forms either the periphery of the cam, or the sides of a 




Ei!fD 

Fig. 2. — Face Cam and Follower 



IFroiee 



groove cut into the face of the cam plate, as illustrated in Figs. 1 
and 2 respectively. Figs. 7 and 9 also show plate cams. 

(g) Heart cams, in which the general form is that which the 
name implies. See Fig. 7. In this type of cam there are two 
distinct symmetrical, lobes, often so laid out as to give uniform 
velocity to the driver. In this case each lobe would be bounded 
by an Archimedean spiral with the ends eased off. 

(h) Frog cam, in which the general form includes several lobes 
more or less irregular, as illustrated, for example at C in Fig. 9. 

(t) Mushroom cam, in which the periphery of a radial or disk 
cam works against a flat surface, usually a circular disk at right 
angles to the cam disk, instead of against a roller, see Fig. 44. 

0) Face cam, also called a Groove, but more properly a Plate 
Groove cam, to distinguish it from the Cylindrical Groove cam, in 
which a groove is cut into the flat face of the cam disk. In 



4 ELEMENTARY CAMS 

this form of cam shown in Fig. 2 the roller has two opposite lines 
of contact, one against each side of the groove, when the roller has 
a snug fit. The plate or disk in which the groove is cut is generally 
circular; but it may be cast to conform with the contour of the 
groove, or it may be built with radial arms supporting the irregular 
grooved rim. In the latter case it lacks resemblance to the face 




Front 
Fig. 3. — Cylindrical Cam and Swinging Follower 



cam, but nevertheless it must, because of the nature of its action, 
be classed with it. The face cam, as ordinarily considered and as 
illustrated in Fig. 2, is better adapted for higher speeds because of 
its more nearly balanced form of construction. Against this, how- 
ever, must be considered one of two disadvantages, either the high 
rubbing velocity of the roller against one side of the groove when 
the roller is a snug fit, or lost motion and noise as the working line 
of contact changes from one side of the groove to the other when 
the roller has a loose fit. The most important advantage of the 
face cam, that of giving positive drive, will be considered in para- 
graph 9. The term groove cam might be applied, with advantage 
in clearness of meaning, to such face cams as are cut or cast on 
non-circular plates. 



DEFINITIONS AND CLASSIFICATION 5 

(k) Wiper cam, which has an oscillating motion, and is con- 
structed usually with a long curved arm in order that it may "wipe'' 
or rub along the plane surface of a long projecting 'Hoe," or follower. 
The wiper cam is used generally to give motion to a follower which 
moves straight up and down as shown from F to F' in Fig. 8. This, 
however, is not essential and the follower may also have a swinging 




Fig. 4. — End Cam and Follower 



motion. The disadvantage of sliding friction, which is inseparable 
from the wiper cam, is balanced to some extent by the fact that 
the very sliding permits, within certain range, of the assignment of 
specified intermediate velocities between the starting and stopping 
points which cannot be obtained with similar forms of cams which 
have pure rolling action. 

(l) Rolling cam, which greatly resembles the wiper cam in 
general appearance, but which is totally different in principle, for 
the curves of the cam and follower surfaces are specially formed so 
as to give pure rolling action between them. The rolling cam is 
specially well adapted to cases where both driver and follower have 
an oscillating motion and where the velocities between the starting 
and stopping points are not important and are not specified. 



6 



ELEMENTARY CAMS 



(m) Yoke cam, a form of radial cam in which all diametral lines 
drawn across the face and through the center of rotation of the 
cam are equal in length. This form of cam permits the use of 
two opposite follower rollers whose centers remain a fixed distance 
apart, to roll simultaneously on oppo&ite sides of the cam, and thus 
give positive motion to the follower. For illustration, see Fig. 9. 





Pig. 5. 



-Conical Cam and Recipeo- 
CATiNG Follower 



Fig. 6. 



-Spherical Cam and Swinging 
Follower 



Yoke cams may be, and frequently are, made of two disks fixed 
side by side, the peripheries being complementary to each other 
and the two rollers of the yoke rolling on their respective cam surfaces, 
as shown in Fig. 56. The advantage of yoke cams is that they 
give positive motion with pure rolling of the follower roller, there 
being contact on only one side of the roller in contradistinction to 
the double contact of th^ roller which exists in face and groove 
cams. 

5. The following specially named cams fall under the general 
classification of side cams. 

These include cams that have been made from blank cylindrical 
bodies by using a rotary end cutter with its axis at right angles 
to the axis of the cylinder and by moving the axis of the rotary 
cutter parallel to the axis of the cylinder while the cylinder rotates. 
A groove of desired depth is thus left in the cylinder, Fig. 3, or the 
end of a cylindrical shell is thus milled to a desired form, Fig. 4. 
A side cam may also be formed by screwing a number of formed 



DEFINITIONS AND CLASSIFICATION 



clamps on to a blank cyl- 
inder, the sides of the 
clamps thus acting as the 
working surface as illus- 
trated in Fig. 11. All 
types of side cams may 
properly be considered 
as derived from blank 
cyHndrical forms, and, 
therefore, the name *' cyl- 
indrical cam'' could be 
regarded as S3rnonymous 
with side cam; but gen- 
eral custom has limited 
the use of the term cyl- 
indrical cam to the '' bar- 
rel" or "drum" type 
mentioned below: 

(n) Cylindkicalcam, 
also called Barrel cam, 
Drum cam, or Cylindrical 
Groove cam,in which the 
groove, cut around the 

cylinder, affords bearing surface to the two opposite sides of the 

follower roller, thus giving positive motion, as illustrated in Fig. 3. 

(o) End cam, in which the working surface has been cut at the 

end of a cylindrical sheel, thus re- 
quiring outside effort such as a 
spring or weight to hold the follower 
roller against the cam surface during 
the return of the follower. An end 
cam is shown in Fig. 4. 

(p) Double end cam, in which 
a projecting twisted thread has been 
left on a cylindrical body, against 
both sides of which separate rollers 
on a follower arm may operate, 
and thus secure positive motion. 
Instead of cutting down a cylinder 
to leave a projecting twisted thread. 
Fig. 8.— Toe and Wiper Cam it may be cast integral with a 




Fig. 7. — Heart Cam and Follower, Sliding Contact 




8 



ELEMENTARY CAMS 



warped plate, as illustrated in Fig. 10, but this in no way changes 
its characteristic action. 

There are a number of names in common use for cams, that 
cover both radial and side cams. Most prominent in this connection 
are those mentioned in paragraphs 6 to 14. 

6. Box CAM, which designates a cam in which the follower roller 
is encased between two walls as in the face cam. Fig. 2, or the cylin- 
drical cam. Fig. 3. Literally, box cams would also include yoke 
cams, in which the yoke would be the "box." Box cams, because of 
their form of construction, give a positive drive in all cases. 

7. Inteknal cam, in which there is only one working surface, 
and this is outside of the pitch surface. The internal cam cor- 
responds to the internal gear wheel in toothed gearing. It may also 
be considered as a face cam with the inside surface of the groove 
removed, thus requiring that the follower roller should always be in 
pressure contact on the outside surface of the groove by means of a 
spring or weight, etc. Under some conditions of structural arrange- 
ments of the cam machine, the internal cam may be used to advan- 
tage where it will give a positive motion to a follower on the opposite 
stroke to that of the periphery cam; and it will also sometimes 




Fig. 9. — Yoke Cam 



permit of a larger roller than the periphery cam, as explained in 
paragraphs 56 and 62. 

8. Offset cam, in which the line of action of the follower^ 
when extended, does not pass through the center of the cam, 
see Fig. 43. 

9. Positive-drive cam is one in which the cam itself drives 
the follower on the return as well as the forward motion. Most 



DEFINITIONS AND CLASSIFICATION 



9 



cams drive only on the forward motion of the follower and depend 
upon gravity or the action of a spring to drive the follower in its 
return motion; such cams are illustrated in Figs. 1, 4, 5, 6, 7, and 8. 
Cams having positive drive, and therefore independent of gravity 
or springs, are illustrated in Figs. 2, 3, 9, and 10. It will be noted 
that positive-drive cams include the face, yoke, cylindrical, and 
double-end types of cams; also that the box cam, although it in- 
cludes some of these, should also be considered as a group name of 
the positive-drive type. 

10. Single-acting and double-acting cams comprise all forms 
of cams, the single-acting ones giving motion only in one direction 
and depending on a spring or gravity to return the follower. Double- 
acting cams have the follower under direct control all the time and 
are the same as positive-drive cams described in the preceding 
paragraph. 

11. Step cams. Cams which give continuous motion to the 



likiin^ 




Front 



Fig. 10. — Double-End Cam 



follower from one end of the stroke to the other are called single- 
step cams. When the follower's motion in either of its two general 
directions is made up of two entirely separate movements it is called 
a double-step cam with reference to that stroke. If three or more 
separate movements are given to the follower while it moves in one 
general direction it is generally referred to as a multiple step cam. 
or as a triple-step, quadruple-step cam, etc. Since a cam may be, 
for example, a double-step cam on the out or working stroke, and 




o o o o 
v<r--v o o <=) 



Front 



Ehd 



Fig. 11. — Barrel Cam 




Front e^td 

Fig. 12. — Adjustable Plate Cam 




Front End 

Fig. 13. — Dog Cam 



DJ£1FINITI0NS AND CLASSIFICATION 11 

a single-step cam on the return stroke, such a cam may be referred 
to as a two-one step cam, always giving the number referring to the 
working stroke first. 

12. Adjustable cam, also called clamp cam, strap cam, dog 
CAM, and carrier CAM, in which specially formed pieces are directly 
bolted or clamped to any of the regular geometrical surfaces, usually 
to either the plane or cylindrical surfaces. In Fig. 12 the clamps 
are shown at C and D fastened to a disk. The cam, considered as a 
whole, belongs to the radial class. In Fig. 13 the clamps are shown 
at C and D, also fastened to a disk, but in this case the clamps, or 
dogs, as they are usually called when used in this way, are so formed 
as to give a sidewise motion to the follower, and therefore this cam 
belongs to the side cam class. In Fig. 11 clamps are shown at C, 
D, £J, and F fastened to a cylinder, and they are shaped to give the 
same action as a regularly formed end-cam in the side-cam class. 
The type of cam illustrated in Fig. 11 is also known as an adjustable 
cylindrical or ''barrel" or ''drum" cam and is very widely used for 
regulating the feeding of the stock, and in operating the turret in 
automatic machines for the manufacture of screws, bolts, ferrules, 
and small pieces generally that are made up in quantities. 

13. Double-mounted or multiple-mounted cams are some- 
times resorted to where several movements can be concentrated 
into small space. This consists merely in placing two or more of 
any of the cam surfaces described in the preceding paragraphs on 
one solid casting or cam body. For example, a face cam, a cylin- 
drical, and an end cam may all be cut on one piece. 

14. Oscillating cams, in which the cam itself turns through a 
fraction of a turn instead of through the entire 360°. While any 
type of cam may be designed to oscillate instead of rotate, it is 
usually the toe-and-wiper and rolling forms of the radial type of 
cam that are known as oscillating cams. With oscillating cams the 
follower may move forth and back on a straight line, or it may 
oscillate also. 

15. Cams falling in the conical class have no special name other 
than the one here used. The spherical cams are sometimes termed 
globe cams. Cams in conical and spherical classes are particularly 
useful in changing direction of motion in close quarters and in 
directions other than at right angles. In both Figs. 5 and 6, end 
action of the cam is shown, but it is apparent that with thicker walls 
on both the cone and the sphere, grooves could be cut in them, 
thus giving positive driving cams in both cases. 



12 



ELEMENTARY CAMS 



16. Summing up the general and special names for cams we 
have in tabular form: 



Cams 









e Periphery 








/ Plate 








g Heart 




Box 




a Radial 


h Frog 




Internal 




or Disk 


i Mushroom 




Offset 
Positive Drive 






j Face or Plate Grooved 








k Toe and Wiper 




Single Acting 
Double Acting 
Step 
Adjustable or 

Strap 
Dog or Carrier 


> J 




I Rolling 

. m Yoke or Duplex 
n Cylindrical, Grooved, Barrel, 


or 


1 


b Side, or 

Cylindrical 


Drum 
End 








p Double End 




Multiple 






- 




Mounted 










Oscillating 




c Conical 








d Spherical 










or Globe 







Definitions of Teems Used in the Solution of Cam Problems 

17. Cam chart. Illustrated in Fig. 14. The chart is a rectangle 
the height of which is equal to the total motion of the follower in 
one direction, and the length equal to the circumference of the pitch 
circle of the cam. The chart length represents 360° and is sub- 




270° 



Equal to length of circumferen«e of pitch circle of cam 

Fig. 14. — Cam Chart 



divided into equal parts marking the 5°, 10^ 



points, or the J^, 



3^ . . . points, or any other convenient subdivision, according to 
the requirements of the problem. On the cam chart are drawn the 
base curve and the pitch line. The former becomes the pitch surface 
of the cam and the latter the pitch circle. 

18. Cam chart diagram. Illustrated in Fig. 15. The cam 
chart diagram is a rectangle, the height of which represents the 
total motion of the follower in one direction. The length of the 
diagram represents the circumference of the pitch circle of the cam. 



DEFINITIONS AND CLASSIFICATION 



13 



In the cam chart diagram the scales for drawing the height and the 
length of the rectangle are totally independent of each other and 
independent also of the scale of the cam drawing. In drawing the 
diagram no scale need be used at all, and the entire chart diagram 
with its base curve and pitch line may be drawn entirely freehand 
with suitable subdivisions marked off entirely "by eye" according 
to the requirements of the problem. The base curve may be drawn 
roughly as a curve or it may be made up of a series of straight lines. 
The cam chart diagram frequently serves all the purposes of the 
cam chart. It saves time, and permits of chart drawings being 



^^^9-- 




90" /^o'' 270° 3 m 

■Represents knqf/i of c/'rcumferencz ofpffch circ/e of cam 

Fig. 15. — Cam Chart Diagram 



made on small available sheets of paper, whereas the more precise 
cam chart often requires large sheets of paper which are usually 
impracticable in academic or exercise problems. 

19. Time charts. Illustrated in Figs. 16 and 17. Time charts 
are the same as cam charts or cam chart diagrams, and are con- 
structed in the same way as described in the two preceding para- 
graphs. The term "time chart," however, is most appropriately 
applied to problems where two or more cams are used in the same 
machine and where their functions are dependent on each other. 



" 


W 1 

SL ^s, ,. 


2JA ~ ~" -^ 




r- ^1^ ""^~" 


'"" 'C^ 


.1 /' W- 


:\ st.:^.. 



0° 90° 180° 270° 360" 

Fig. 16. — Time Chart Diagram, Base Curves Superposed 

The time chart permits of allowances being made for avoiding 
possible interference of the several moving parts, and for the desired 
timing of relative motions for each part. The time chart contains 
two or more base curves according to the number of cams used. 
When the base curves are superposed as in Fig. 16, the time chart 
consists of a single rectangle whose height is equal to the greatest 



14 



ELEMENTAKY CAMS 



follower motion. The superposing of curves and lines often leads 
to confusion and error, and it is better, in general, that the time 
chart should consist of a series of charts or rectangles all of the 
same length and one directly under the other as in Fig. 17. Where 
there are many base curves it is desirable to separate the rectangles 



e 


































< 


^ 


? 


y 


/ 


/ 






N 


s 


V 


s 


\ 


\ 


\ 


s 


\ 


V 






z' 


/ 


/ 


/ 


y 














' 












« 
















C 


i 


I 


y 




s 


\ 


V 










































t 










N. 


s 


^ 


•s 


V 


^ 


■N. 








L 




















S2^ 


5 


y 


y 


</ 


^ 


^ 







0° 90= 180° 270° 360" 

Fig. 17. — Time Chart Diagram, Base Curves Separated 



by a small space to avoid any possibility of confusion due to different 
base curves running together. In many cases the term ''time chart 
diagram,'^ or "timing diagram," will be more appropriate than "time 
chart" in just the same way as the cam chart diagram is more ap- 
propriate than the cam chart. 

20. Base curve. Illustrated in Fig. 14. A base curve is made 
up of a series of smooth continuous curves, or a combination of 
curves and straight lines, which represent the motion of the follower, 
and which run in a wave-like form across the entire length of the 
cam chart or diagram. The base curve of the cam chart becomes 
the yiicln surface of the cam. 

21. Base line. Illustrated in Fig. 15. A base Hne is made up 
of a series of inclined straight lines, or a series of inclined and hori- 
zontal lines, in consecutive order, which zigzag across the entire 
length of the chart. The base line when used on the cam chart 
indicates the exact motion of the follower, but when used on a cam 
chart diagram it is merely a time-saving substitute for the drawing 
of the base curve. The base line of the cam chart diagram represents 
the pitch surface of the cam. 

22. Names of base curves or base lines in common use, 
see Figs. 18 and 19: 

1. Straight line 4. Parabola. 

2. Straight-line combination 5. Elliptical curve. 

3. Crank curve. 



DEFINITIONS AND CLASSIFICATION 



15 



23. Pitch line. Illustrated in Fig. 14. A pitch line is a 
horizontal line drawn on the cam chart or diagram, and it becomes 
the pitch circle of the cam. The position, or elevation, of the pitch 
line on the chart varies according to the base curve which is specified, 
and according to the data of the problem. For cams which give a 




Fig. 18. — Comparison of Base Curves in Common Use Showing Varying Degrees 
OF Maximum Slope When Drawn in Same Chart Length 

continuous motion to the follower during its entire stroke, or throw, 
the pitch line will pass through the point on the base curve which 
has the greatest slope, starting from the bottom of the chart. This 
does not apply to all possible base curves, but it does apply to all 




Fig. 19. 



-Comparison of Base Curves in Common Use Showing Uniform Maximum 
Slope of 30® When Drawn in Charts of Varying Length 



those mentioned in the preceding paragraph, a minor exception 
being made of the crank curve which will be referred to in para- 
graph 34. When the cam causes the follower to move through its 
total stroke in two or more separate steps the position of the pitch 
line on the chart must be specially found as will be explained in 
problem 5. 



16 ELEMENTARY CAMS 

24. Pitch circle. Illustrated in Fig. 20. A pitch circle is drawn 
with the center of rotation of the cam as a center, and its circumfer- 
ence is equal to the cam chart length. Its characteristic is that it 
passes through that point A, Fig. 20, of the pitch surface of the cam 
where the cam has its greatest side pressure against the follower. 
This applies to all cams in which the center of the follower roller 
moves in a straight radial line. For other motions of the follower 
roller, and for flat-faced followers, the pitch circle must be specially- 
considered, as will be explained in some of the problems covering 
these types. 

25. Pitch surface. Illustrated in Fig. 20. The pitch surface 
of a cam is the theoretical boundary of the cam that is first laid 
down in constructing the cam. When the follower has a V-shaped 
edge, as at D in Fig. 7, the pitch surface coincides with the working 
surface of the cam. When the follower has roller contact, as in 
Fig. 20, the pitch surface passes through the axis of the roller and 
the working or actual surface of the cam is parallel to the pitch 
surface and a distance from it equal to the radius of the roller. 

26. Working surface. Illustrated in Fig. 20. The working 
surface of the cam is the surface with which the follower is in actual 
contact. It limits the working size and weight of cam. For exact 
compliance with a given set of cam data, the cam has only one 
theoretical size which is bounded by the pitch surface, but the 
working size may be anjd^hing within wide limits which depend on 
the radius of the follower roller and the necessary diameter of the 
cam shaft. 

The working surface is found by taking a compass set to the 
radius of the roller and striking a series of arcs whose centers are 
on the pitch surface. Such a series of arcs is shown in Fig. 20 with 
their centers at B, A, etc. The curve which is an envelope to 
these arcs is the working surface. 

27. Pitch point of follower. Illustrated in Fig. 20. The 
pitch point of the follower is that point fixed on the follower rod or 
arm which is always in theoretical contact with the pitch surface 
of the cam. If the follower has a sharp V-edge the pitch point is 
the edge itself. If the follower has a roller end, the pitch point 
is the axis of the roller. The pitch point is constantly changing its 
position from C to D as the follower moves up and down. 

28. Pressure angle. Illustrated in Fig. 20. The pressure 
angle is the angle whose vertex is at the pitch point of the follower 
in its successive positions and whose sides are the direction 



DEFINITIONS AND CLASSIFICATION 



17 



of motion of the pitch point and the normal to the pitch 
surface. 

Pressure angles exist when the surface of the cam presses sidewise 
against the follower; they cause bending in the follower arm and 
side pressure in the follower guide and in the bearings. The pres- 



'Maximum Pressure 




Fig. 20. — Showing Names or Sukfaces, Lines, and Points of a Cam 



sure angle is constantly varying in all cams as the follower moves 
up and down, except where a logarithmic spiral is used. In assign- 
ing cam problems the maximum permissible pressure angle is usually 
given. In Fig. 20 the pressure angle is zero at C, it is equal to a 
at B, and is a maximum at A, 

29. Formula for size of cam for a given maximum pressure 
ANGLE. The radius of the pitch circle of the cam may be found 
directly by the formula: 



'-"-?x/xi 



or, 



= 57.3 



h X 



= .159 



hf 
h 

e 

hf 
e 



X/>^2^ 



(1) 



(2) 



18 



ELEMENTARY CAMS 



in which, r = radius of pitch circle of cam. 
h = distance traveled by follower. 
/ = factor for a given maximum pressure angle. 
h = angle, in degrees, turned by cam while follower moves 

distance h. 
e = angle, in fraction of revolution, turned by cam while 

follower moves distance h. 

30. Cam factors for maximum pressure angle. The factors, 
or value of /, for various maximum pressure angles for cams using 
the several base curves in common use are: 

Table of Cam Factors 



Name of Base Curve 


Maximum Pressure Angle 


AND Values 


OF/ 


20° 


30° 


40° 


50° 


60° 


Straight line .... 


2.75 
3.10 
4.32 
5.50 
6.25 


1.73 

2.27 
2.72 
3.46 
3.95 


1.19 
1.92 

1.87 
2.38 
2.75 


.84 
1.77 
1.32 
1.68 
1.95 


.58 


Straight-line combination* . . . 
Crank curve " . . . . . 


1.73 
.91 


Parabola 


1.15 


Elliptical curve t 


1.35 







These factors, for 30°, are illustrated in Fig. 19 where each of 
the base curves is given such a length, in terms of the height, that 
they will all have the same maximum slope. The values given in 
this table are also shown, graphically, in Fig. 21, thus enabling one 
to find the proper cam factor for any intermediate pressure angle 
between 20° and 60°. 



* For case where easing off radius equals follower's motion. 

t For case where ratio of horizontal to vertical axes of ellipse is 7 to 4. 



DEFINITIONS AND CLASSIFICATION 



19 



, B L DQ F 

60 1 \n \ 


C^SV 


V V^-s 


A ^.-V^ 


56 ^^ ^^ns^ 


^° t N NCv 


4 5i^^ 


\. ^^CvS 


W Xl^ ^^ 


40 -& "^i s s|.. 


V^ A \ AjV- 


uz N ^*^sr 


^Sr v^j. ^FnTT 


Xt-JW&o ^ ^^tj 


oS gi^^^v^S it 


vi^ s^ ^^^ 


' ^^:M '^s. ^^^^ 


S^t ^v. ^^^^^ 


^^. -^^^. ^^^^"-^ 





•4 /'T 



If 



1 3 4 3J5 

Cam Factors 

Fig. 21. — Chabt Showing Relation Between Pressure Angles and Cam Factors 
FOR the Ordinary Base Curves 



SECTION IL— METHOD OF CONSTRUCTION OF BASE 
CURVES IN COMMON USE 

31. Detail construction of base curves. The method of 
constructing the several base curves for a rise of one unit of the 
follower will be explained in the succeeding paragraphs. The curves 
will be constructed to give a pressure angle of 30° by selecting factors 
from the 30° column in the table in the preceding paragraph. Should 
the base curve for any other pressure angle be desired the factor 
should be taken from the corresponding column. 

32. Straight-line base. Fig. 22. Lay oil A B equal to the 
follower motion, which will be taken as 1 unit in these illustra- 
tions. Multiply this by the factor 1.73 from paragraph 30, and 
lay off the distance A R equal to it. Complete the parallelogram 
and draw the diagonal. This will be the straight line base and the 




Fig. 22. — Straight Base Line 



Fig. 2.3. — Straight-Line Combination Curve 



angle RAC will be 30°. A R will be the pitch Hne. These base 
lines and curves are laid off from right to left so that they may be 
used in a natural manner later on in laying out the cam so that 
it will turn in a right-handed or clockwise direction. 

The straight-line base gives abrupt starting and stopping velocities 
at the beginning and end of the stroke and causes actual shock in 
the follower arm. The velocity of the follower during the stroke is 
constant. The acceleration at starting and retardation at stopping is 
infinite and is zero during the stroke. 

33. Straight-line combination curve. Fig. 23. Construct 
the rectangle with a height of 1 unit and a length of 2.27 units. 
With B and R as centers draw the arcs A E and C N, and draw a 
straight line E N tangent to them. The angle FEN will then equal 
30° and the line A C will be a base curve made up of arcs and a 

20 



CONSTRUCTION OF BASE CURVES IN COMMON USE 



21 



straight line combined to form a smooth curve. D F will be the 
pitch line. 

The straight-line combination curve, being rounded off at the 
ends, does not give actual shock to the follower at starting and stop- 
ping, but it does give a more sudden action than any of the base 
curves which follow, and the maximum acceleration and retardation 
values are comparatively larger. 

34. Crank Curve. Fig. 24. Construct the rectangle. Draw 
the semicircle R G C and divide it into any number of equal parts. 
Six parts are best for practice work for this curve, but in general 
in practical work the greater the number of divisions the more 
accurate will be the curve and the smoother the action of the cam. 



bTT 




FlG. 24. — CkANK CUKVi 



The six equal divisions of the semicircle are readily obtained by 
taking G as a center and F C as a radius and striking arcs at 1 and 
5, then with R and C as centers mark the points 2 and 4 respectively. 
Divide the length of the chart into six equal parts, as at H, I, E, etc. 
From these points drop vertical lines, and from the corresponding 
divisions on the semicircle draw horizontal lines, giving intersecting 
points, as at K, on the desired crank curve. The tangent to the 
curve at E will then make an angle of 30° with the line E F. The 
pitch line will he D F. 

When the crank curve is transferred from the chart to the cam 
it gives an angle which is a fraction of a degree greater than 30° 
at the point E on the cam in practical cases. This is not enough 
greater to warrant the special computations and drawing that would 
be necessary to be exact. Therefore the method of laying out the 
crank curve and the pitch line, as given above, will be adhered to 
in this elementary consideration of cam work, because of its 
simplicity. 

The crank curve gives a slightly irregular increasing velocity 
to the follower from the beginning to the middle of its stroke; then 
a decreasing velocity in reverse order to the end of the stroke. The 



22 



ELEMENTARY CAMS 



acceleration diminishes to zero at tiie middle of the stroke and then 
increases to the end. The maximum acceleration and retardation 
values are much less than for the straight-line combination curve, 
and are only a little greater than for the parabola. 

35. Paeabola. Fig. 25. Construct the rectangle. Draw the 
straight line R S in any direction and lay off on it sixteen equal 
divisions to any scale. From the sixteenth division draw a line to F, 
the middle point of the chart; draw other hues parallel to this 
through the points 9, 4, and 1, thus dividing the distance R F into 
four unequal parts which are to each other, in order, as 1, 3, 5, and 
7. From these division points draw horizontal lines, and from H^ 
I, and J drop vertical lines. The intersecting points, as at K, 




Fig. 25. — Parabola 



will be on the desired parabola. The points H, I, and J divide the 
distance D E into four equal parts. 

The parabola gives a uniformly increasing velocity from the 
beginning to the middle of the stroke; then a uniformlj^ decreasing 
velocity to the end. The acceleration of the follower is constant 
during the first half of the stroke and the retardation is constant 
during the last half. The acceleration and retardation values are 
equal and are less than the maximum value of any of the other base 
curves. This means that the direct effort required to turn a positive- 
acting parabola cam is less than for anj^ other type of positive cam. 

36. To better understand the smooth action given by the cam 
using this curve, consider, 1st, D H aus a time unit during which the 
follower rises one space unit; 2d, ^ / as an equal time unit during 
which the follower rises three space units; 3d, I J as the time unit 
during which the follower rises five space units, etc. Inasmuch as 
the follower travels two units further in each succeeding time unit, 
it gains a velocity of two imits in each time unit, and this is uniform 
acceleration. 

The distance from F to C would be divided the same as from 
F to R and points on the part of the cm-ve from E to C similarly 



CONSTRUCTION OF BASE CURVES IN COMMON USE 



23 



located. This curve will be identical with E A, but in reverse order, 
and will give uniform retardation. The tangent to the curve A C 
at the point E will make an angle of 30° with E F, and D F will 
be the pitch line. 

Eight construction points were taken in developing the curve 
A C. Eight points will be sufficient for beginners for practice work 




Fig. 26. — Elliptical Curve 



and later six points may be used. When using six points only nine 
equal divisions should be laid out on the line R S, the remaining 
constniction being the same as described above, except that D E 
should be divided into three parts instead of four. In practical work 
many more construction points should be used for accuracy and 
smooth cam action. 

37. Elliptical curve. Fig. 26. Draw rectangle A B C R. 



Draw semi-elUpse making F G equal to -j F C. 



To draw the ellipse, 



take a strip of paper with a straight edge and mark fine lines at 
P, T, and S, Fig. 26a, making P T = C F and P S = G F. Move 
the strip of paper so that S will always be on 
the fine R C, and T on'the line F G; P will then 
describe the path of the ellipse. Having the semi- 
ellipse, divide the part R G, Fig. 26, into four 
equal arcs as at i, 2, 3. This is quickest done 
by setting the small dividers to a small space of 
any value and stepfping off the distance from R 
to G. Suppose that there are 18.8 steps. Set 
down this number and divide it into four parts, 
giving 4.7, 9.4, and 14.1. Then again step off the 
arc from R to G with the same setting of the dividers, marking the 
points that are at 4.7, 9.4, and 14.1 steps. The compass setting 
being small, the fractional part of it can be estimated with all prac- 
tical precision. Divide D E into four equal parts as at H, I, J. 
Draw vertical lines from these points and horizontal lines from the 




E\ 

Fig. 2Ga. — Showing 
Method of Draw- 
ing Semi-Ellipse 



24 ELEMENTARY CAMS 

corresponding points at 1, 2^ and S. The intersections, as at Ky 
will give a series of points on the elliptical base curve. The curve 
E C is similar to A ^ but in reverse order. The tangent to the curve 
at E makes an angle of 30° with E F, and D F is, the pitch line. 

The elhptical base curve gives slower starting and stopping 
velocities to the follower than any of the other curves, but the velocity- 
is higher at the center of the stroke. The acceleration is variable and 
increases to the middle of the stroke, where its maximum value is 
greater than that of the crank curve but less than that of the straight- 
line combination curve. The retardation values decrease in reverse 
order to the end of the stroke. 



SECTION III.— CAM PROBLEMS AND EXERCISE 
PROBLEMS 

38. Problem 1. Empirical design. Required a radial cam 
that will operate a V-edge follower: 

(a) Up 3 units while the cam turns 90°. 

(b) Down 2 " " " " " 60°. 

(c) Dwell " " " " 120°. 

(d) Down 1 unit " " " " 90°. 

39. Applying the simplest process for laying out cams, it is only 
necessary, in starting, to assume a minimum radius C D, Fig. 27, for 




Fig. 27. — Empirical Design of Cam for Data in Problem 1, V-Edge Follower 

the cam, and then lay off the given or total distance of 3 units as 
at D B. The assigned angle of 90° is next laid off as at Z) C Di and 
the point Di marked so as to be 3 units further out than D. Any 
desired curve is then drawn through the points D and Di and part 
of the cam layout is completed. The same operations are repeated 
for obtaining the points D2 and Da and the entire cam is finished. 
If the follower had roller contact instead of V-edge contact, a 

25 



26 



ELEMENTARY CAMS 



minimum radius C D, Fig. 28, would be assumed as in the previous 
case, and D would be taken as the center of the roller. The closed 
curve D, Da, Di . . . would be obtained as before and another closed 
curve E, El . . . would be drawn parallel to it at a distance equal 




PiQ. 28 — Empirical Design of Cam for Data in Problem 1, Eoller Follower 



to the assumed radius of the roller. The latter closed curve would 
be the actual outline of the cam. 

The closed curve E Ei . . . would be known as the working sur- 
face and the curve D Di . . .as the pitch surface of the cam. In 
Fig. 27 the pitch and working surfaces coincide because the follower 
has a V-edge. 

40. Cams are sometimes designed with no more labor than that 
entailed in the previous preliminary problem. And it may be added 
that where one has had a sufficient experience good practical results 
may be obtained by following only this simple method. 

The method of cam construction described above, however, does 
not enable the cam builder or designer to hold in control the velocity 
or acceleration of the follower rod D G as it moves up its 3 units; 
nor does it enable him to know the variable and maximum side pres- 
sures which exist between the follower rod and the bearing or guide 



CAM PROBLEMS AND EXERCISE PROBLEMS 



27 



F, Fig. 27, as the rod moves up. In order that these things may 
be known, this preHminary problem will now be redrawn with ad- 
ditional specifications. 

41. Problem 2. Technical desigtst. Required a radial cam that 
will operate a roller follower: 

(a) Up 3 units while the cam turns 90°. 



(b) 
(c) 
(d) 



Down 2 " 
Dwell 
Down 1 unit 



60°. 

120°. 

90°. 



(e) The follower, in all its motions, shall start with uniform 
acceleration and stop with uniform retardation. 

(f) The angle of side pressure of the follower rod against the 
guide shall not exceed 40°. 

Items (a), (b), (c), and (d) are the same as in Problem 1. 

42. Inasmuch as this problem is given at this place simply to 
show that velocity and acceleration and side pressure can always 
be controlled with very little additional labor beyond that necessary 
for the simple layout shown in Fig. 28, the full explanations of the 
formula and figures used will not be given here. They will be taken 
up in their proper order in subsequent paragraphs. For this problem 
the only necessary computation is: 



4.55 = Radius of pitch circle 



X 



r~h 






1 

B 






=^-i 


-4 


1 ' 


'fl 


;^--^ 




i 




t- 


NT 



C H, Fig. 29. 

The reference letters, /i, /, 
and b are defined in paragraph 
29. Lay off C /^ in Fig. 29, and 
then lay off the follower motion 
of 3 units equally distributed on 
each side of H, as Sit H B and 
H D. Divide D H into nine 
equal parts and take the first, 
fourth, and ninth parts; do like- 
wise with B H. Divide the 90° 
angle B C Di into six equal 
parts by radial lines as shown, 
and swing each of the six di- 
vision points between D and B 
around until they meet succes- pig. 29.-technical design of cam for data 

, ^, . 1. 1 1. IN Problem 2, Drawn to Same Scale 

sively the six radial hues, as fiq. 28 



28 



ELEMENTARY CAMS 




A curve through the intersecting points will be the pitch surface of 
the cam, as shown by the dash-and-dot curve D Hi Di. . . . 

The working surface will be EEi 
. . . which is found as described 
in paragraph 26. 

The pitch surface Di D2 is 
obtained in the same way as 
D Di was found. The curve 
Z>2 D3 is an arc of a circle, and 
the curve D3 D is found in the 
same manner as D Z)i. 

43. Advantages of the 

TECHNICAL DESIGN. With the 

cam constructed as above the 
follower will start to move with 
the same characteristic motion 
as has a falling body starting 

Fig. 29.— (DupUcate) Technical Design of f j-Qm rest, and the follower will 
Cam for Dat4 in Problem 2, Drawn to i • i i 

Same Scale as Fig. 28 be stopped With the Same gen- 

tle motion in reverse order. It 
will be definitely known also that the greatest side pressure 
of the cam against the follower is at an angle of 40° as specified, 
and that this pressure will occur when Hi of the pitch surface of the 
cam is at H, or when the roller is in contact with the working 
surface at H2. Where the cam form is assumed as in Fig. 28, nothing 
is known positively of the starting and stopping velocities of the 
follower. Further, as may be found by trial, the maximum angle of 
pressure of the cam against the rod runs up to 47° in Fig. 28, as 
shown at D4. The minimum radius of the cam in Fig. 28 was taken 
equal to that in Fig. 29 for comparison. 

44. The two previous problems have been given as brief exercises 
without going into all the detail necessary to a full understanding, 
in order to give an idea of the method of producing cams on a scientific 
basis. In the problems which will follow, the several steps in building 
cams of various types will be explained. In many of the problems 
the same data will be used so that comparisons of different forms 
of cams which produce the same results may be made. 

45. Problem 3. Single-step radial cam, pressure angle 
EQUAL ON BOTH STROKES. Required a single-step radial cam in 
which the center of the follower roller moves in a radial line. The 
maximum pressure angle to be 30°, and the follower to move: 



CAM PROBLEMS AND EXERCISE PROBLEMS 



29 




S>' 



(b) Down 3 
acceleration and 



(a) Up 3 units in 90° with uniform acceleration 
and retardation. 

units in 90° with uniform 
retardation. 

(c) At rest for 180° with uniform acceleration 
and retardation. 

46. The first step in the solution is to determine 
the total length of the cam chart for a parabola 
chart curve and for a 30° maximum pressure angle. 
From the table, paragraph 30, the factor for this 
case is found to be 3.46. Since the travel of the 
follower is 3 units in }/i revolution, the total length 
of chart will be 3 X 3.46 X 4 = 41.52, which, 
therefore, is the length of the chart A A' in Fig. 30. 
This length represents the 360° of the cam. Lay- 
off A TF equal to 90°, according to item (a) in 
the data. Construct the parabolic curve A E C. 
Completing the entire chart, the base curve is 
found to he AC M N A\ The next step is to find 
the radius of the pitch circle. The circumference 
of this circle is equal to the length of the pitch 

41.52 
Its radius is, therefore, equal to ~ — = 



i « line DD\ 



6.61, and this value is laid off at D, Fig. 31, and 
the pitch circle D F Q W drawn. The quadrant 
D F is divided into the same number of parts 
as D F in Fig. 30. The vertical construction lines 
H Hi, II 1, J Ji . . . in Fig. 30 now become the 
radial lines correspondingly lettered in Fig. 31, 
and the pitch surface is drawn through the points 
A Hill J I. . . . The positions of maximum pres- 
sure are shown at E and Q; at all other points it 
will be less. The working surface B G R P is 
found by assuming a radius A B for the roller, 
and by striking a series of arcs as shown at //•:, 
I2, J2 . . . with the points Hi,Ii,Ji ... as cen- 
ters, and then drawing the working curve tangent 
to these arcs. With the same specifications for 
the up and down motions of the follower, as 
given by items (a) and (b) in the data, this type 
of cam will be symmetrical about the line Y C. 



30 



ELEMENTARY CAMS 




Fig. 31. — Problem 3, Cam Laid Out fkom Cam Chart 




FiQ.32. — Problem 3, Cam Laid Out Independently of Cam Chart 



CAM PROBLEMS AND EXERCISE PROBLEMS 31 

47. Omission of cam chart. When the relation between pres- 
sure angle, chart base and pitch lines, and cam pitch and surface 
lines is understood and fixed in mind, the actual drawing of the 
chart for the graphical construction of simple cams and particularly 
of single-step cams may be omitted with full confidence when the 
elementary base curves are used. For example, the problem in the 
previous paragraph is shown completely worked out in Fig. 32 
without any reference whatever to the chart of Fig. 30. The radius 
D of the pitch circle, Fig. 32, is obtained directly from the formula, 

r = 57.3 -r given in paragraph 29. Substituting the data as given 

3 X 3.46 
in the previous paragraph, r = 57.3 ^ — = 6.61 and is laid off 

at D 0. The assigned motion of the follower is laid off symmetri- 
cally on both sides of the pitch point D, as at A V, and the distances 
A D and V D are divided into the desired number of imequal parts, 
as at 1, 4, 9, 16. The quadrant D F is divided into the same number 
of equal parts as sit H, I, J . . . and indefinite radial construction 
lines drawn through the points. Circular construction arcs are 
next drawn through the points 1, 4, 9 . . . until they intersect the 
radial lines, thus obtaining points Hi, 7i, Ji . . . on the cam pitch 
surface. In general, a neater construction is obtained by omitting 
the full length of the construction arcs, as from F to C . . . and 
simply drawing short portions of the arc at the intersecting radial 
lines as shown in the lower left-hand quadrant between C and M. 

48. Exercise problem 3a. Required a single-step radial cam 
in which the center of the follower roller moves in a radial fine. 
The maximum pressure angle to be 40°, and the follower to move: 

(a) Out 6 units in 135° on the crank curve. 

(b) In 6 " " 135° " '' 

(c) At rest for 90°. 

49. Problem 4. Single-step radial cam, pressure angles 
UNEQUAL on THE TWO STROKES. Required a single-step radial cam 
in which the center of the follower moves in a radial line. The 
maximum pressure angle not to exceed 30° on the outstroke nor 50° 
on the return stroke, and the follower to move: 

(a) Out 2 units in /ig revolution -on the crank curve. 

(b) In 2 '' '' %Q '' '' " " " 

(c) At rest for ^2 revolution. 



32 



ELEMENTARY CAMS 



50. The diameter of pitch circle of the cam that will be necessary 
to fulfil the requirements on the outstroke will be: 
2 X 2.72 X 16 



da = 



3.14 X 5 



= 5.54 units, or from formula paragraph 29. 



. = ■159 ^X^^X^^ = 2.77, 



and the diameter of pitch circle required for the instroke will be 
2 X 1.32 X 16 



3.14 X 3 



= 4.48 units. 



Inasmuch as there can be only one pitch circle for a cam, the 
largest one resulting from the several specifications must be used. 
In this problem then the diameter S D of the pitch circle in Fig. 33 




PiQ. 33. — Problem 4, Maximum Pressure Angle Different on the Two Strokes 

equals 5.54 units. The follower's motion of two imits is laid out 
at A F and the pitch surface AE C M N constructed. The working 
surface of the cam B KG, etc., is then drawn. Since a larger diameter 
of pitch circle had to be used for the return stroke than the require- 



CAM PROBLEMS AND EXERCISE PROBLEMS 33 

ments called for, it follows that the pressure angle will not reach 
50° on that stroke, and it may be of some interest to determine what 
the maximum pressure angle on the return stroke will be. Sub- 

stituting the diameter used, 5.54, in the formula d = — and solving 

for /, / is found to be equal to 1.63. From the chart in Fig. 21 it 
is shown that a factor of 1.63 for the crank curve corresponds to a 
maximum pressure angle of nearly 44°, and this angle may be drawn 
in its proper position at Q in Fig. 33. 

51. Exercise problem 4a. Required a single-step radial cam 
in which the center of the follower roller moves in a radial line. The 
maximum pressure not to exceed 30° on the up stroke nor 40° on the 
down stroke, and the follower to move : 

(a) Up 3 units in 135° on the parabola curve. 

(b) At rest for 45°. 

(c) Down 3 units in 90° on the parabola curve. 

(d) At rest for 90°. 

52. Pressure angle increases as pitch size of cam decreases. 
This is illustrated in Fig. 34, where the large pitch cam represented 
hy D, D2 . . . gives exactly the same motion to a follower as the 
small pitch cam d, d^. . . . It will be noted that the pressure angle 
for the large cam, at the start, is H D G, while for the small cam it 
is increased to hdg. Likewise the maximum pressure angle for the 
large cam, when the follower is near the end of its stroke, is 61, 
while for the small cam the maximum pressure angle is b, which is 
larger than 61. From these observations it may be said, in general, 
that the larger the pitch surface of the cam the smaller will be the 
pressure angle. The size of the roller has no effect whatever on 
the pressure angle. Two cams of the same pitch size may be of 
totally different actual sizes for the same work, one cam having 
a large roller and the other a small roller. Therefore it is important 
to remember that, in general, the pressure angle may be regulated 
by changing the size of the pitch surface only and not the working 
surface. 

53. Change of pressure angle in passing from chart to 
CAM. The circumference of the pitch circle of the cam, it will be 
recalled, is equal to the length of the pitch line on the chart. It 
will also be remembered that the pitch line may be at various heights 
on the chart, paragraph 23. It is now important to consider: 

1st. That the pressure angle at the pitch circle on the cam must 
be the same as the pressure angle at the pitch line on the chart. 



34 



ELEMENTARY CAMS 



2d. That the pressure angle at any point on the pitch surface 
of the cam outside of the pitch circle will be less than the pressure 
angle of the corresponding point on the base curve of the cam chart. 

3d. That the pressure angle at any point on the pitch surface 




D^y^ -" Z>5 

Fig. 34. — Showing Relation Between Pressure Angle and Size of Pitch Cam 

of the cam inside of the pitch circle will be greater than the pressure 
angle of the corresponding point on the base curve of the cam chart. 

These statements, which are theoretically true for nearly all cases, 
and practically so for all other cases where the usual base curves 
are employed, are demonstrated in the following paragraph. 

54. Cam considered as a bent chart. Consider that the cam 
itself is the cam chart bent in its own plane so that the pitch line 



CAM PROBLEMS AND EXERCISE PROBLEMS 



35 



becomes the pitch circle. Then the line D D', Fig. 30, becomes 
the circle D F OWy Fig. 31 ; the line V V' is stretched to become 
the circle V C SY, and the straight line A M A' is compressed to 
become the circle A M A. This means, in a general way, that 
the rectangle DVV' £>', Fig. 30, is so distorted that if an original 
diagonal had been drawn from D to F' it would have an increased 
length and a decreasing slant after the bending had taken place. 
With a decreasing slant of the pitch surface the pressure angle will 
decrease. Likewise, a diagonal drawn from D' to A in the original 
rectangular chart would be decreased in length and would have an 
increasing slant, and the pressure angle would be increasing toward 
A. This is illustrated in detail in Figs. 35 and 36. 

55. Base line angles, before and after bending. The pres- 
sure angle of 30° at E in Fig. 35 is reduced to 23° in Fig. 36, and the 




Fig. 35. — Section of Cam Chart Be- 
FORB Bending 



Fio. 36. — Section of Cam Chart After 
Bending, BC Constant in Bot^h Figures 



30° at D are increased to 41°. Fig. 35 represents a cam chart with 
a straight base line D E, and Fig. 36 is a corresponding cam sector 
with D E as the pitch surface. If B (7, Fig. 35, is taken as the 
pitch line, B C, Fig. 36, will be part of the pitch circle. The uniform 
pressure angle of 30° from A to E, Fig. 35, will grow smaller beyond 
A in Fig. 36 for the reason that the radial components of the tan- 
gential triangles remain constant, as illustrated at L M, while the 
tangential components grow longer as illustrated at A iV and E L, 
which are respectively equal to the arcs A Y and E Li. Con- 
sequently, the angles grow smaller from the angle N A P to L E M. 
Similarly it may be shown that they grow larger from N A P to 
QDR. 

56. Limiting size of follower roller. The radius of the 
follower roller may be equal to, but in general should be less than 



36 



ELEMENTARY CAMS 



the shortest radius of curvature of the pitch surface, when measured 
on the working-surface side. If the radius of the roller is not so 
taken, the follower, when put in service, will not have the motion 
for which it was designed. 

57. Case 1. Radius of roller equal to radius of curvature 
OF PITCH CAM. In Fig. Z7, A B E F A is the pitch surface of a cam. 




FiQ. 37. — Limiting Size of Follower Roller 



CAM PROBLEMS AND EXERCISE PROBLEMS 37 

G A is the radius of curvature at A and A G is the radius of the 
roller. In this case both radii are equal and the working surface 
has a sharp edge at G. 

58. Case II. Radius of roller greater than radius of 
CURVATURE OF PITCH CAM. From B to C, Fig. 37, the radius of 
curvature of the pitch surface is H B, which is less than the roller 
radius. In this case the working surface will be undercut at / in 
generating the cam, and if the cam is built the center of the roller 
will mark the path Bi Ji Ci instead oi B J C, and the follower will 
fail to move the desired distance by the amount J Ji. 

59. Special application of case II. Effect of an angle in 
THE pitch surface OUTLINE. This is illustrated at R F Qin Fig. 37, 
and is a special application of Case II, in which the radius of cur- 
vature of the cam's pitch surface is reduced to zero. Undercutting 
is here illustrated by considering that a cutter, represented by the 
dash circular arc, is moving with its center on the pitch surface 
arc E F. It then cuts the working surface M S. As the center of 
the cutter is moved from F toward A, the part N S oi the working 
surface which was previously formed is now cut away, leaving 
the sharp edge N on which the follower roller will turn when the 
cam is placed in operation. The center of the follower roller will 
then move in the path R T Q instead oi R F Q, and the follower will 
fall short of the desired motion by the amount T F. 

60. Case III. Radius of roller less than radius of curva- 
ture OF PITCH CAM. From D to E, Fig. 37, the radius of curvature 
of the pitch surface is i^ JD, which is greater than the roller radius. 
In this case, which is the practical one, although close to the limit, 
a smooth curved working surface is provided for the roller from 
LtoM. 

61. Radius of roller not affected by radius of curvature 
ON non-working side. From Ci to D, Fig. 37, the radius of curva- 
ture of the pitch surface is less than the radius of the roller, but 
this short radius is not on the working side of the pitch surface, 
and therefore the roller will roll on the surface / L while its center 
travels on the pitch curve CiD. 

62. Rollers for positive-drive cams. When the largest roller 
for a positive or double-acting cam is being determined the radius 
of curvature on both sides the pitch-surface curve must be con- 
sidered and the smallest radius used. For example, in Fig. 37, if 
A J E T A were the pitch surface for a double-acting cam, N C 
would be the maximum roller radius, whereas H J would be 



38 



ELEMENTARY CAMS 



the maximum radius if it were for an external single-acting 
cam. 

63. Radius of curvature of non-circular arcs. In illustrat- 
ing the above cases the pitch surface was assumed as being made 
up of straight lines and arcs of circles in order to show more effec- 
tively and more simply the limits of action in each instance. Where 
the pitch surface contains curves of constantly var3dng curvature, 
and they generally do in practice, the shortest radius of curvature 
of the pitch surface may be found with all necessary accuracy by 
trial with the compass, using finally that radius whose circular arc 
agrees for a small distance with the irregularly curved arc. For 
example, in Fig. 38, let G H D J B be a portion of a pitch surface 




Fig. 38. — Limiting Size of Follower Roller Working on Non-Circular Cam Curves 



made up of non-circular arcs. The shortest radius of curvature 
on both sides is found, by trial, to be F H. The center F is marked 
and the osculatory arc X H Z drawn in. Then H F is the largest 
possible radius of roller for a double-acting cam, and with this 
roller the working surfaces will he V F T W smd Vi Fi Ti Wi. 



CAM PROBLEMS AND EXERCISE PROBLEMS 39 

If a larger roller is used, with a radius D R, for example, the 
working surfaces of the groove will he S E and Pi Ki iVi, and the 
new pitch surface, after cutting the cam, will he G C D L B, if the 
roller is kept always in contact with the inner surface of the groove. 
If it is kept always in contact with the outer surface of the groove, 
the original pitch surface will be changed to G C H DiJ B. In 
either case the original desired follower motion is not obtained if the 
roller is too large, and if a positive-drive cam is run with the larger 
roller the follower's motion will be indeterminate, the center of 
the roller having any possible position between C D L and C H J L. 

64. Problem 5. Double-step radial cam. Required a double- 
step radial cam in which the center of the follower roller moves in 
a radial hne. The maximum pressure angle to be 30°, and the 
follower to move: 

(a) Up 4 units in 3^ revolution on the crank curve. 

(b) At rest for 3^ revolution. 

(c) Up 4 units in y^ revolution on the parabola curve. 

(d) Down 2 units in J^ revolution on the elliptical curve. 

(e) At rest for f/s revolution. 

(f) Down 6 units in J^ revolution on the parabola curve. 

65. In Problem 3 there are only two motion assignments, (a) and 
(b), in the data, and they were the same except for direction. Con- 
sequently only one computation was necessary. When two or more 
dissimilar assignments are made in the data, as in the present problem, 
it is advisable to make a computation for the length of the chart 
diagram for each motion specification, as follows: 

(a) 4 X 2.72 X 8 = 87.04, which is the length of chart and of 

the pitch circle circumference = 
13.86 pitch circle radius. 

(c) 4 X 3.46 X 8 = 110.72, which is the length of chart and pitch 

circle circumference = 17.62 pitch 
circle radius. 

(d) 2 X 3.95 X 8 = 63.20, which is the length of chart and pitch 

circle circumference = 10.06 pitch 
circle radius. 
(f) 6 X 3.46 X 4 = 83.04, which is the length of chart and pitch 

circle circumference = 13.22 pitch 

circle radius. 

Inasmuch as there is a different length of chart and a different 

pitch line for each item in the data one can not tell which pitch line 

to take without some preliminary computation. For this purpose 



40 



ELEMENTARY CAMS 



a chart diagram is well adapted, as follows: Construct a rectangle, 
Fig. 39, with a height A T equal to the total motion of the follower 
in one direction, 8 units in this case. Make the length A A' of 
rectangle any convenient value entirely independent of any of the 
values computed above and label this according to the longest chart 
length as computed above. Lay off straight lines to represent the 
component parts of the base curve as assigned in the data and label 
them as shown at A C, C B, B H, etc. Draw the several pitch lines 
as at F D, J I, etc. 

66. For general procedure, consider the pitch line which passes 
through the point calUng for the longest chart length. This will 
be the pitch Hne J I passing through G, Fig. 39, which calls for a 
chart length of 110.72 and a pitch radius of 17.62. If G is to be at 



T 


, 








H 












? 


U 

00 

i 
D 

4>' 






M 4 


.,^^ 


A^ 






: 


^. 


6 




^^j. 


^A 


I'i 


u 


^o^ 


/ 


M, t L 


r 


B 


e 


' s 






t 


C i 


Q 






i 


. F 


\ . 


z 


p 


I 






1 


a' 




= 




E\ 




'^ %..... H ^ % K J^ 


























' 





Fig. 39. — Problem 5, Cam Chart Diagram for Double-Step Cam 



a radius of 17.62 in the cam, E will be at a radius of 17.62 - 4 = 
13.62. But from computation (a) it is seen the E must be at a 
radius of 13.86. Therefore, if the radius of cam pitch circle is 
retained at 17.62, the trial pitch line, J /, on the chart diagram will 
have to be lowered, 13.86 - 13.62 = .24, giving the new pitch line 
U U', If the line U U' now becomes the pitch circle the point E 
will be at 17.62 - 3.76 = 13.86, just as called for in computation 
(a), and the pressure angle will be 30° at the point E on the cam. 

The other critical points at P and K must also be tested with 
respect to the proposed pitch line, U U'. With this pitch line the 
point P will be 2.76 inside of the pitch circle, or at a radius of 17.62 - 
2.76 = 14.86. This is safe, as the computed radius for P was only 
13.22 according to item (f). The point K is also safe, for it will be 
at a radius of 17.62 + 1.24 = 17.86, whereas a radius of only 10.06 
is required. 

67. The cam may now be drawn by constructing the true cam 
chart as in Fig. 40, which is lettered the same as Fig. 39, and plotting 
the cam from it as in Fig. 41. The pitch line U U' of Fig. 40 be- 



CAM PROBLEMS AND EXERCISE PROBLEMS 



41 



ifj;^ 



-♦I q> ^r-< 



^1 b o^ -T\ ^ 



comes the pitch circle, having a radius U in 
Fig. 41, and the ordinates of Fig. 40 become 
the radial measuring lines in Fig. 41. Or the 
cam may be drawn directly, without the use of a 
cam chart, as indicated in Fig. 42, where the 
pitch circle U' is first drawn with a radius of 
17.62. The assigned angles are then laid down 
and the several pitch curves, such as A' E C, are 
constructed at the proper radial distances as de- 
termined in Fig. 39 and as illustrated for one 
case at E Ei (3.76) in Fig. 42. 

68. Determination of maximum pressure 
angle for each of the curves making up a 
MULTIPLE-STEP CAM. If it is desired to know the 
exact pressure angle at P, Fig. 41, it may be readily 
determined by making the value of r = (17.62 — 

2.76)= 14.86 in the formula, r = .159—^ and 

solving for /, the notation being the same as given 
in paragraph 29. 

14.86 
^ .159X6X4 



= 3.J 



Consulting the chart of cam factors in Fig. 21, 
it is found that a factor of 3.89, when applied to 
the parabola chart curve, shows a cam pressure 
angle of about 27°, which is under the assigned 
limit, and therefore need not be further consid- 
ered. In a similar manner the pressure angle at 
G and K on the cam may be computed if desired, 
the former being a small fraction of a degree 
under 30° and the latter something less than 20°, 
the reading running off the chart. 

69. Exercise problem 5a. Required a double- 
step radial periphery cam in which the center of 
the follower roller moves in a radial line. The 
maximum pressure angle to be 30° and the follower 
to move: 

(a) Out 5 units in 150°, witn uniform accelera- 
tion and retardation. 

(b) In 2 units in 30° on the crank curve. 



42 ELEMENTARY CAMS 

(c) At rest for 60°. 

(d) In 3 units in 120° on the elliptical curve. 

70. Problem 6. Cam with offset roller follower. Re- 
quired a single-step radial periphery cam in which the center of the 
follower roller moves forth and back in a straight hne which does 
not pass through the center of rotation of the cam. The maximum 
pressure angle when the follower is at the bottom of its stroke is 
to be 30°, and the follower is to move: 

(a) Up 3 units in 90° on the parabola curve. 

(b) Down 3 '' '' 90° '' " 

(c) At rest for 180°. 

71. Problems of this nature are totally different, both in pressure- 
angle action and in methods of construction, from the preceding 
ones. As may be noted in the data, it is required that the pressure 
angle, when the follower is at rest at the bottom of its stroke, shall be 30°. 
It will appear presently that the pressure angle, when the follower 
is in motion, may be zero or even negative on one of the strokes in 
this form of cam. It will also be shown that the maximum pressure 
angle during the follower motion cannot be assigned in advance and 
obtained in any practical manner. From the above it follows that 
the offset radial cam has a peculiar advantage in keeping considerable 
side pressure off the follower guides during the time that the follower 
is moving in one direction, although at the bottom of the stroke the 
pressure angle may have any desired value, and during the period 
of motion in the opposite direction the pressure angle will reach a 
maximum value much larger than the assigned angle at the bottom 
of the stroke. 

72. The method of construction for the offset roller cam is illus- 
trated in Fig. 43. The diameter of the pitch circle, U F' ST, is com- 

puted as before by the formula, d = 114.6 -r-, and found to be 6.61 

units. An angle equal to the assigned pressure angle is then laid 
off at U U\ U being parallel to the direction of motion of the 
follower. Draw a line, DW, parallel to U and so located that it 
has an intercept D A between the pitch circle and the inclined line 
equal to one-half the travel of the follower. This may be done by 
trial, or graphically, as shown by the dotted-line construction which 
is drawn at Y' X A' instead of at Y to avoid complication of con- 
struction lines. The angle U Y' equals the angle U Y. Y' X, 
parallel to U 0, is drawn equal to one-half the stroke. An arc X A , 



CAM PROBLEMS AND EXERCISE PROBLEMS 



43 



parallel to the arc F' U, is drawn through X by using C/ as a radius 
and Z as a center, where Z equals Y' X. A circular arc through 
A' with as a center will intersect U' in the desired point A. The 
point A will then be the lowest point of the stroke, D will be the 
center of the stroke, and W the radius of the construction circle. 




Pig. 41. — Problem 5, Double-Step Cam Constructed from Cam Chart 




Fig. 42. — Problem 5, Double-Step Cam Constructed Without Use of Cam Chart 



The distance A F is equal to the assigned 3 units of motion, and 
the divisions 1, 4) ^ • • • are made according to the requirements of 
the parabola curve. The assigned 90° is laid off on the construction 
circle Sit W F and divided into a number of equal arcs at H, /, 
J . . . corresponding to the number of divisions at A V, eight being 
used in the present example. Tangents to the construction circle, 



44 ELEMENTARY CAMS 

such as H Hi, I Ii, J Ji . . . are then drawn at H^ I, J . . . and the 
distances Wl, W4, ^y9 . . . laid off on these tangents, thus gi^dng 
the points Hi, h, Ji . . . on the pitch surface of the cam. Or these 
latter points may be obtained by swinging arcs through 1, 4, 9 . . . 
about as a center, until they meet the respective tangents at Hi, 
h, Ji. . . . 

73. An examination of the pressure angles for a cam with an 
offset follower shows that during the up stroke the pressure angles 
are very small, being, in fact, negative from Ji to Ki, Fig. 43, and, 
when measured, the average pressure angle for the working or up 
stroke is between 6 and 7 degrees in this problem; although on the 
down or return stroke it reaches an average of between 37° and 38° 
and a maximmn of 46° near Q\ In this class of problem the com- 
putation for diameter of pitch circle serves merety as a guide in deter- 
mining a size that will give a small cam and a small average pressure 
angle on the working stroke. If the diameter of the pitch circle is 
arbitrarily taken either larger or smaller than the value, as above 
computed, or if other base curves are used, the negative pressure 
angles at Ji, Ei, and Ki may disappear entirely; which would be an 
advantage where it is desired to have pressure on the follower guides 
on one side only. 

74. It has doubtless been observed that there is a decided lack 
of sjTnmetry in this form of cam, even though the data are similar 
for both strokes of the foUower. This is illustrated in Fig. 43, where 
the portion A C of the pitch surface for the outstroke is quite different 
from the portion C M. It is also characteristic of this form of cam 
that the pitch and working curves each embrace either a smaller or 
a larger angle than the assigned angle for a given stroke of the follower, 
as shown by the angle A C being less, and the angle C M being 
greater, than the assigned 90°. This, of course, is due to the fact 
that when C has traveled 90° to V the line C will have passed the 
original zero line A oi the pitch curve and will be in the position 
V. Therefore, the cam angle for one stroke of the follower will be 
less than the assigned angle by the amount of the angle included by 
V A; for the other stroke it will be greater than the assigned 
angle by the same amount. 

75. Exercise problem 6a. Required a single-step radial periph- 
ery cam in which the center of the follower roller moves forth and 
back in a straight line which does not pass through the center of 
rotation of the cam. The maximum pressure angle when the follower 
is at the bottom of its stroke is to be 40°, and the follower is to move: 



CAM PROBLEMS AND EXERCISE PROBLEMS 45 

(a) Out 6 units in 135° on the crank curve. 

(b) In 6 " " 135° " '' " 

(c) Rest for 90°. 

In this problem, only the initial pressure angle at the bottom of 
the stroke need be shown; the pressure angles at other positions, 
such as are shown in Fig. 43 at Hi, h, may be omitted. 




Fig. 43. — Problem 6, Cam with Offset Roller Follower 

76. Problem 7. Cam with flat surface follower, — Mush- 
room CAM. Required a radial periphery cam to operate an offset 
follower which has, a flat surface instead of a roller. The follower 
to move: 

(a) Up 3 units in 90° on the parabola base. 

(b) Down 3 '' " 90° '' " 

(c) At rest for 180°. 

77. This type of cam is known also as the mushroom cam. Flat 



46 



ELEMENTARY CAMS 



surface followers may be offset as shown in the side and top views 
in Fig. 44, where the center line A^" Y'' of the follower spindle is set 
the distance P" 0" in front of the center of the cam plate. In this 
case there will be a part sliding and part rolling of the cam on the 
follower and the follower will turn about its own axis, N" F," as it 




Fig. 44. — Problem 7, Cam with Flat Surface Follower — Mushroom Cam 



is being raised and lowered. When the follower is not offset, i.e., 
when the center hne 0" N" is placed in hne with M" P", the action 
will be all sHding and^there will be no turning of the follower spindle 
on its axis. In this case there will be localized wear on the follower, 
while in the former case the wear will be more widely distributed 
over the follower surface. In both cases the construction is the 
same and is explained in the following paragraph. 

78. In cam followers having flat surfaces perpendicular to the 
line of action, the line of pressure is M" Q" and is parallel to the 
line of action of the follower, instead of being inclined to it as in 
the case of cams having roller followers. Because of this character- 
istic action the ordinary pressure-angle factors do not apply in 



CAM PROBLEMS AND EXERCISE PROBLEMS 47 

cams of this class in computing or obtaining the diameter of the 
pitch circle D F S Tj and this circle may be assumed. In some 
cases a fair guide for the size of this circle may be obtained by using 

h f 

the regular formula, d = 114.6 -r-, for diameter of pitch circle, as- 
suming the 30° pressure angle factor. Solving, d is found to equal 
6.61, and is laid off at D. The assigned three units of motion are 
then laid off, one-half on each side of D, as at A and F. The assigned 
90° are next laid off at A /^^ and divided into the desired number 
of construction parts, four being used in this case, as at A, D2, 
. . . The distance A 7 is also divided into four parts, A H and V K 
being each equal to 1 unit and H D and K D equal to 3 units. Only 
four divisions are taken in this case to avoid confusion of lines in 
the illustration, but in student problems 6 or 8 points should be 
taken, and in practical work 12 to 24 divisions should be used. 
The first division point, H, is now revolved to meet the first radial 
division line Di, thus giving the point Hi, where a line Hi E is 
drawn perpendicular to Hi 0. This line Hi E represents the bottom 
of the follower disk A C with reference to the cam when the cam 
has turned through the angle A Hi. The points D2 and Ki are 
obtained in the same manner as was Hi and corresponding perpendic- 
ulars are drawn, as at D2 D4 and Ki K2. As smooth a curve as possible 
is now drawn tangent to these perpendiculars and the points of 
tangency marked as at H2, D4, and K2. This smooth curve, A G, 
is the working surface of the cam. 

79. The size of the follower must also be determined. The most 
satisfactory way of doing this is to find, first, the locus, or path, 
of the line of contact between the periphery of the cam and the 
follower disk. This is obtained by considering that when Hi is at 
H, the point of tangency H2 is at H3, the length H H3 being equal to 
Hi H2. Likewise, when D2 is at D, D^ is at D5, and the same for the 
other points of tangency. The dash fine curve through the points 
A H^D^Kz . . . i^ the locus of contact between the cam and the 
follower. The point L is the extreme point of this curve and if the 
follower were not offset, the length of an ordinary toe or flat extension 
of the follower would have to be at least equal to N' X/ If the 
follower is offset, say by the amount N' M' {= N'^ M"), the radius 
of the disk will have to be at least equal to N' L', and the extreme 
line of contact will be L' j'. The other extreme line of contact will 
be a similar line through L'", and the area of the flat disk which will 



48 



ELEMENTARY CAMS 



be subject to wear will be the annular surface between the peripher}^ 
and the dashline circle whose radius is iV' A\ As to the wear on the 
cam itself, there would be pure shding of the curved surface A G 
on the flat surface AX ii the follower were not offset. With an 
offset follower there is an effective turning radius equal to the offset 




Fig. 44. — (Duplicate). Problem 7, Cam with Flat Surface Follower- 
Mushroom Cam 



N^ M' tending to rotate the follower about its axis N" 7", and this 
changes the action of the cam on the follower entirely by causing 
part rolling and part sliding. 

80. The pressure angle in this form of cam must be considered 
differently from cams which operate against rollers. In roller 
follower cams it is the angle between the normal to the cam surface 
and the line of action of the follower that determines the side pres- 
sure on the bearings, whereas in flat surface followers it is the distance 
that the line of contact is away from the line of action that determines 
it. This distance varies constantly, and in the illustration in Fig. 
44 the limits of variation are M' N' and Q' N\ These are, in reality, 
lever arms on which the pressure acts to produce a turning moment 



CAM PROBLEMS AND EXERCISE PROBLEMS 49 

which must be resisted by the follower guides. Since there can be 
no pure rolling action between the cam and follower in constructions 
of this type, there is nothing to be gained in this particular by a 
large offset. On the contrary, there is much to be lost, due to the 
large turning moment on the follower rod. A fair guide as to the 
offset would be to keep the angle formed by the center line Y" 0" 
of the follower motion and the line Y" M" or Y" Q" joining the 
center of the bearing with the midpoint of the line of contact, to 
within, say, 30°, or any other maximum value that circumstance 
might warrant. The angle here defined might be termed the pres- 
sure angle in this type of cam. The minimum pressure angle, 
Isl" Y" M" ^ is seen in its true size, while the maximum pressure 
angle as projected at V" Y" Q" must be revolved about V Y" 
as an axis until U" Q" equals iV' Q' , when it will appear in its true 
size as at U" Y" Q"\ 

81. Limited use of cams with flat surface followers. Cams 
with followers of this type are not well adapted, in general, for cases 
in which the follower must have specified velocities during its stroke. 
If the follower is required only to move from one end of its stroke 
to the other in a given period of time, independently of all inter- 
mediate velocities, this form of construction may be readily applied. 
The principal difficulties to be met in the building of these cams, 
when the intermediate velocities are specified, are, first, the large 
time angles necessary for a desired follower motion, or, second, a 
comparatively large cam. The cause of these difficulties may be 
pointed out in Fig. 44, where it may be seen that the construction 
point, Ki, n^ght have been so much further out radially that the 
perpendicular line, Ki Ki, would have passed to the left of E and it 
would have been impossible to draw the smooth cam curve A G 
tangent successively to all the perpendiculars. The limiting prac- 
tical case appears when any three successive construction lines 
meet in a point, in which event the cam will have a sharp edge and 
be subject to excessive wear at that point. This subject is further 
considered in paragraph 106. 

82. If one is not limited in the time, or angle, in which the follower 
must do its work; or, if not limited in the size of the cam, this form 
of construction may be used for any set of velocity values so long 
as they produce a working surface which always curves outward 
or which has an edge which points outward. 

83. Exercise problem 7a. Required a radial peiiphcry cam 
to operate an offset follower which has a flat surface pcrpen- 



50 ELEMENTARY CAMS 

dicular to the line of motion instead of a roller, the follower 
to move: 

(a) Up 3 units in 90° on the crank curve. 

(b) Down 3 " " 90° " " 

(c) Rest for 180°. 

Take cam disk to be one unit thick and the follower offset equal 
to two units measured from center of cam disk. Find and mark 
the locus of contact, also the size of the follower disk and the area 
of follower surface subject to wear. 

84. Cams for swinging follower arms. In the previous prob- 
lems the motion of the center of the follower roller has been in a 
straight line. When the center of the roller moves in a curve a 
different method of construction is used to advantage. Cams with 
swinging followers are illustrated in Figs. 45 and 46, the arc of 
swing A V oi the follower having its extremities on a radial line in 
the former illustration; and on an arc which, continued, passes 
through the center of the cam in the latter illustration. These two 
forms of construction, although apparently differing in only a slight 
detail, give quite different results and each has its own particular 
field of usefulness. A comparison of the results will be given in 
paragraph 95 after a problem in each case has been worked out. 

85. Problem 8. Cam with swinging follower arm, roller 
CONTACT — Extremities of swinging arc on radial line. Re- 
quired a radial periphery cam to operate a roller follower where the 
follower arm swings about a pivot and where the two extreme posi- 
tions of the center of the roller lie on a radial line. The chord of 
the swinging arc of the roller center is to be 4 units and the length 
of the follower arm 8 units. The follower arm to swing: 

(a) Out full distance on 90° on parabola curve. 

(b) In '' '' " 90° " crank 

(c) And to remain at rest for 180°. 

86. A different method of construction from any thus far em- 
ployed is used in problems of this kind because it gives the simplest 
and most accurate layout for the pitch surface. Briefly, the method 
to be used consists in revolving the follower through the 360° around 
the cam while the cam remains stationary, and drawing the follower 
in a number of its phases while on the way around. One of the 
phases is represented in full by the dash lines Cio F2 Y3 in Fig. 45. 

87. The angle which causes pressure against the follower bear- 
ings is also different in this form of cam from any of the others. An 
inspection of Fig. 45 will show that, in general, the normal line of 



CAM PROBLEMS AND EXERCISE PROBLEMS 51 

pressure, AV at A, between the cam surface and the roller is noi at 
right angles to the position of the follower arm, and, therefore, 
that the resultant total pressure has a component along the arm, 
tending to place it in compression and throwing a corresponding 
pressure on the follower bearing at C The pressure angle at A is 
shown by —a, the minus sign indicating compression in the swinging 
arm. When Ki is at K the pressure angle will be +c, the plus sign 
indicating tension in the follower arm. A disadvantage of the sign 
changing from + to — , etc., is that as soon as the bearings wear 
there will be noise at that point. 

88. The detail for the construction of problem 8 is taken up 
by computing the diameter of the pitch circle first, as in previous 
problems. This computation, however, serves only as a guide, for 
the assigned pressure angle will be both increased and decreased by 
amounts depending on the radius of the follower arm and the char- 
acteristics of the base curve which is used. For computing the 
pitch circle then, an assigned pressure angle factor for 30° will be 
assumed in the expectation that the final maximum angle will not 

4 X 3.46 
exceed 40°. From formula 1, paragraph 29, d = 114.6 — ^ — 

4 X 2.72 
= 17.62 for the parabola curve; and d = 114.6 — ^ — = 13.86 

for the crank curve assignment. The radius of the pitch circle is 
thus found to be 8.81 units. 

89. Having determined the radius D, Fig. 45, for the pitch circle, 
the given chord of 4 units is laid off with equal parts on each side 
of D, thus locating the ends of the swinging arc A F on the radial 
line D as required. With A and V as centers and a radius of 
8 units for the length of the follower arm, strike arcs which will 
intersect at C and give the fixed center for the follower arm. The 
arc A V, showing the path of the center of the follower roller is now 
drawn. 

90. Points on the pitch surface A V1A2F are found, in brief, 
by revolving the arm C A around 0, swinging it a proper amount 
on its center C as it revolves. In detail this is accomplished by lay- 
ing off the arc C Ce equal to 90°, and dividing it into a number of 
equal parts, say six. Divide the arc A J into three unequal parts, 
as at H and /, for the parabola curve. Lay off the points L and K 
in the same way. Then with C A as a radius and with Ci, C2 . . . 
as centers draw the arcs passing through Hi, /i . . . . Again, with 



52 ELEMENTARY CAMS 

as a center, swing arcs through H, I . . . until they meet the 
arcs already constructed. The intersections of these arcs, as at 
Hi, 7i, Ji . . . will be the points on the desired pitch surface AVi. 
The determination of the pitch surface for the crank curve is found 
by laying off the second 90° assignment from C& to C^ and dividing 
it into six parts. The arc Ai Fi is divided by projecting the points 
U W . . . oi the crank circle to the points C/i Wi on the arc. The 
constructions for the points U2, W2 . . . are the same as for the 
previous part of the pitch surface, as described above. 

91. Exercise problem 8a. Required a radial periphery cam 
to operate a roller follower where the follower arm swings about a 
pivot and the two extremities of the swinging arc lie on a radial 
line. The 30° pressure angle factor to be used in computing the 
pitch circle radius. The chord of the swinging arc to be 3 units, 
the arm 9 units long, and to : 

(a) Swing out in J^ revolution on the crank curve base. 

(b) Remain at rest for }/s revolution. 

(c) Swing in in 3^ revolution on the parabola base. 

92. Problem 9. Cam with swinging follower arm, roller 
CONTACT — Swinging arc, continued, passes through center of 
CAM. Required a radial periphery cam to operate a roller follower 
where the follower arm swings about a pivot, and where the center 
of the follower roller moves on an arc which, continued, passes 
through the center of the cam. The chord of the swinging arc of 
the roller center is 4 units and the length of the follower arm 10 
units. The follower arm to swing: 

(a) Out full distance in 90° on parabola curve. 

(b) In " " " 90° " crank 

(c) To remain at rest for 180°. 

93. The procedure for this problem is the same as for Problem 8 
in all respects except the layout of the arc of swing for the center 
of the follower roller. The pitch circle is drawn with radius J, 
Fig. 46. 

With the center of the cam and the pitch point J as centers 
draw arcs which intersect at C, the radius being equal to the length 
of the follower arm. Lay off J A and J V equal to each other and 
so that a chord drawn from A to V equals the four units assigned. 
A bent rocker, A C ^, is introduced in Fig. 46 simply to change the 
direction of motion. 

94. Exercise problem 9a. Required a radial periphery cam to 
operate a roller follower where the follower arm swings about a pivot, 



CAM PROBLEMS AND EXERCISE PROBLEMS 



53 



and where the center of the follower roller moves on an arc which, 
continued, passes through the center of rotation of the cam. Take 
the length of follower arm as 12 units and its angle of swing 30°. 
Required that the follower arm: 

(a) Swing out full distance in % revolution, on crank curve. 

(b) Remain at rest 3^ revolution. 

(c) Swing in full distance in % revolution, on crank base. 




Fig. 45.^Problem 8, Cam with Swinging Follower Arm, Roller Contact — Extremi- 
ties OF Swinging Arc on Radial Line 



95. Effect of location of swinging follower arm relatively 
TO THE CAM. When the swinging follower arm is mounted so that 
the extremities of the arc of travel of roller center are on a radial 
line, as in Problem 8, the pressure angles on the out and in strokes 
will be approximately the same. When the follower roller center 



54 



ELEMENTARY CAMS 



moves on an arc which, continued, passes through the center of 
the cam, as in Problem 9, the pressure angle will be larger, on the 
average, on the one stroke than on the other. Consequently, the 
type shown in Problem 8 would have an advantage where equal 
amounts of work were to be done on both strokes, and the type 




Fig. 46. — Problem 9, Cam with Swinging Follower Arm, Roller Contact — Swinging 
Arc, Continued, Passes Through Center of Cam 



shown in Problem 9 would be of advantage where heavy work was 
to be done on one stroke only. Either the out or in stroke may 
be selected for heavy work, according to the position taken for the 
center C or G of the swinging arm, Fig. 46, the direction of turning 
of the cam being the same. In many cases the type shown in 
Problem 9 allows the pressure angle to be maintained on one of the 
strokes so that there is pressure in only one direction on the shaft C. 



CAM PROBLEMS AND EXERCISE PROBLEMS 55 

Cams operate smoother when "rimning off" than when "running 
on." A cam is said to be "running off" when the point of contact 
on the working surface of the cam is moving away from the fixed 
center of the swinging follower arm. A cam of the type illustrated 
in Problem 8 will have an axis of symmetry where the same data 
are assigned for the out and in stroke, whereas the cam illustrated 
in Problem 9 will be quite unsjonmetrical for same data. 

96. Positive-drive face cams. The pitch surfaces for face 
cams are laid out in exactly the same manner as pitch surfaces for 
radial periphery cams. The only additional feature is that a working 
surface is drawn to touch each side of the roller. 

97. Problem 10. Face cam with swinging follower. Con- 
struct a face cam for a swinging follower arm, roller contact. Arm 
to be 12 units long and to swing through 30°. Required that the 
arm shall: 

(a) Swing full out on the combination curve while the cam makes 
5^ revolution. 

(b) Swing full in on the combination curve while the cam makes 
^ revolution. 

98. In order to compute the radius of the pitch circle it is neces- 
sary to find the travel, or the approxi- 
mate travel, of the center of the follower 
roller. This is graphically done by mak- 
ing a separate sketch, as in Fig. 47, 
drawing the angle X Y Z equal to 30°, 
drawing the arc Z X with a radius of 12 
units, and measuring the chord ZX, 
which is found to be 6.2 units. Or, this ^'!:;,fZ^^^'Z'^^Zl^'^^^^ 

^ ' URE OF Travel of Point on 

value may be found trigonometrically, swwiging arm 

without any drawing, by taking 12 X 2 

sin 15° = 6.2. The radius of the pitch circle will then be: 

6.2 X 2.27 X |- X 3-j^ X -i- = 6.0 units. 

99. To construct the cam, the value just found is laid off at J, 
Fig. 48, and the pitch circle drawn. With the combination curve 
a cam chart, a partial one at least, must be drawn. To do this with 
least effort, select any point J' in line with the pitch point J and 
draw the line J' V at the given pressure angle, 30° in this case, 
until it is 6.2 units long. With 7' as a center, draw arc J' A' and 
also draw a tangent to it at J' and produce it to S/ where R aS' equals 




56 



ELEMENTARY CAMS 



one-half A' V. The curve A' J' S' will be one-half of the desired- 
base curve and will be sufficient to proceed with the construction of 
the cam. Divide the pitch line, 0-4, of the chart into four equal 
parts and draw verticals so locating H\ /^ K\ . . . Project these 
points to ^, 7, jK^ . . . on the arc of travel of the center A of the 
roller. This construction will give practically a uniform swinging 




Fig. 48. — Problem 10, Face Cam with Swinging Follower 



velocity to the follower arm through twice the angle measured by 
the arc from J to S. Theoretically, the curve A' S' should be con- 
structed on the cylindrical surface A S instead of on its projected 
plane surface. It is, however, unnecessary to go into the detail 
of construction which this would involve because the difference in 
results between it and the more direct process explained above 
would be too small in practical cases to be measured by the thickness 
of the ordinary pencil line. 

With thfe points H, I, K . . . obtained as above, the remainder 
of the construction is the same in detail as described in connection 
with Problem 8. The reference letters are the same in both figures. 
The cam plate, in the face of which the groove for the roller is cut, 
is made circular in its boundary in order to give better balance and 
appearance. 



CAM PROBLEMS AND EXERCISE PROBLEMS 57 

100. Exercise problem 10a. Required a face cam for a swinging 
follower arm, roller contact. Arm to be 10 units long. Center of 
roller to swing through an arc whose chord is 4 units, and this arc, 
when continued, to pass through center of cam. The arm to: 

(a) Swing to the right on combination curve while cam turns 
180°. 

(b) Swing to the left on combination curve while cam turns 180°. 

101. Problem 11. Cam with swinging follower arm, sliding 
SURFACE CONTACT. Required a radial periphery cam to operate a 
swinging arm having a construction radius of 9 units. Sliding sur- 
face contact between cam and follower. The arm to: 

(a) Swing up 4 units on the crank curve base while the cam 
turns 120°. 

(b) Swing down 4 units on the crank curve base while the cam 
turns 120°. 

(c) Remain at rest for 120°. 

102. This type of cam and follower is illustrated in Fig. 49. The 
line of pressure between cam and follower is always normal to the 
follower surface and consequently there is no component of pressure 
in the bearing at C due to pressure angle. This cam is, therefore, 
independent of a pitch circle based on pressure angle, and the pitch 
circle may be taken any size. Where one has no special guide in 
assuming a starting size for the cam, the usual computation for 
pitch circle for a 30° pressure angle may give good average results. 
According to this, the pitch radius D will be, 

4 X 2.72 X 3 X 3-J1 X 2 = 5.2 units 

AV equals 4 units and A C equals 9 units. The point A is 
taken, for construction purposes, as a point on the follower arm 
where the angular velocity of the arm is measured. It will be 
at the points H, I, J ... on the arc A F at the end of equal suc- 
ceeding intervals of time. 

103. The method of constructing the cam in this problem is 
identical with the method used in Problem 8 in so far as the 
follower arm is swung around the cam, and its position with respect 
to the cam center at equal time intervals is drawn. The departure 
from the method of Problem 8 consists in drawing the cam outline 
as an envelope to these follower-arm positions. For example, in 
Fig. 49, at the end of the third time interval the pivot C has been 
revolved to C3 and the point A of the follower arm has moved out 



58 



ELEMENTARY CAMS 



to Ji. The point /i is found at the intersection of two arcs, one 
obtained with C A as a radius and C3 as a center, and the other 
with J as a radius and as a center. 

When a number of positions of the follower arm, such as C3 Ji, 
have been obtained, the smoothest possible curve is drawn tangent 
successively to each of them, and this curve is the working surface 




Fig. 49. — Problem 11, Cam with Swinging Follower Arm, Sliding Contact 

of the cam. This curve is tangent to C3 Ji at M^ and if the distance 
Cs M is laid off at C Mi, the point Mi will be the actual point of 
tangency between the cam surface and follower arm when the arm 
is halfway through its swing, or when A is at J. Similarly when 
C9 is at C the point of tangency between cam and follower arm will 
be at A^i. 

104. The locus of the point of contact between the cam and 
follower, relatively to the frame of the machine, is shown by the 



CAM PROBLEMS AND EXERCISE PROBLEMS 



59 



dash closed curve through Mi and Ni. By drawing arcs tangent 
at the extremities of this dash curve, using C as a center in both 
cases, the points F and G on the follower surface are obtained and 
the distance F G will be the part of the follower exposed to wear 
from the rubbing of the cam. This part of the follower arm may be 
designed with a shoe, as indicated, which may be replaced when 
worn. 

105. It should be specially noted that the shortest radius of the 
cam is not A, but B. The point B is found by drawing a per- 
pendicular to C G through 0. 

The very decided lack of symmetry should also be noted, the 
curve B L being used to lift the arm, and the curve L E to lower 
the arm, the swinging velocities of the arm being the same in both 
directions. 

106. Data limited for followers with sliding surface con- 
tact. The data for this type of cam construction are extremely 
limited when the swinging velocity of the arm is assigned. The 
limitations are that the working surface of the cam must be drawn 
tangent to every construction line in succession, and that it must be 
convex externally at all points. In most arbitrary assignments of 
data the construction line through Cg, for example, would intersect 
the line through C7 before it cut the line through Cg. In this case 
it would be impossible to draw a smooth working curve tangent, 
successively, to the lines through C7, Cs, and C9. This is illustrated 
more clearly in Fig. 51 and will be more evident after the hmiting 
case is described. 

The limiting case for flat surface followers with sliding contact 
occurs where three or more of the construction lines meet in a point, 
as at N in Fig. 50. In this case the working surface of the cam 




Fig. 50. — Limiting Case for Straight Edge Fig. 51. — Impossible Case for Str.\ight 
Follower with Sliding Contact Edge Follower with Sliding Contact 



60 



ELEMENTARY CAMS 



would have a sharp edge. In this type of cam it is necessary to 
use more construction Knes than in other types, because it is pos- 
sible to have the construction lines so far apart that such a case as 
is shown in Fig. 51 might not evidence itself at all. For example, 
if the distance Cg C7 were the unit space for construction lines, in- 
stead of C9 Cs, the smooth convex curve F N L could be drawn 
tangent to lines through Cg, C7 . . . without the error showing 
itself. 

107. If it is required of this cam only that it shall swing a follower 
arm through a given angle in a given time, without regard to the 




c, ^1 



Fig. 50. — (Duplicate.) Limiting Case for 
Straightedge Follower with Sliding 
Contact 



Fig. 51. — (Duplicate.) Impossible Case 
FOR Straight Edge Follower with 
Sliding Contact 



intermediate velocities of the arm, it may be as widely used as any 
other type of cam. In this case only the innermost and outermost 
positions of the arm would be drawn, as at C A, Ce Fi, and C12 E, 
Fig. 49, and a smooth convex curve drawn tangent to these lines. 
Such construction, however, might give an irregular or jerky motion 
to the follower. Whether it did or not could be readily determined 
by laying off a number of equal divisions, as at Ci, C2 . . . C12; 
drawing lines, such as C3 /i, tangent to the assumed smooth convex 
working surface; and revolving C3 Ji back to C J. After doing 
this with other construction lines a series of points, such as H, I, J 
. . . would be determined and the spaces between them would 
represent the distances traveled by A on the follower arm during 
successive equal intervals of time. 

108. Exercise problem 11a. Required a radial periphery cam 
for a swinging follower arm, sliding surface contact. Arm to be 
10 units long to the point which is used to measure the angular 
velocity, and this point to move through an arc which is measured 
by a chord of 4 units. The arm is to: 



CAM PROBLEMS AND EXERCISE PROBLEMS 61 

(a) Swing full out with uniform acceleration and retardation 
while the cam turns % revolution. 

(b) Swing in with the same angular motion in % revolution. 

(c) Remain stationary for 3^ revolution of the cam. 

109. Toe and wiper cams. In this form of cam construction 
the cam or ''wiper" OC, Fig. 52, oscillates or swings back and 
forth through an angle of 120° or less, instead of rotating con- 
tinuously the full 360° as it does in all cams thus far considered. 
The follower or "toe" A TT is usually a narrow flat strip resting on 
the curved periphery of the cam, and moving straight up and down. 
There is sliding action between the wiper and the toe. 

110. Problem 12. Toe and wiper cam. Required a wiper cam 
to operate a flat toe follower which shall move: 

(a) Up 4 units with uniform acceleration all the way while the 
cam turns counterclockwise 45° with uniform angular velocity. 

(b) Down 4 units with uniform retardation all the way while 
the cam turns clockwise 45° with uniform angular velocity. 

111. The detail of construction for this class of problem is iden- 
tical with that described for the mushroom cam in Problem 7, 
it being observed that the two cams differ only in that the mush- 
room cam turns through the full 360° instead of 45° as in this problem, 
and the mushroom follower is circular instead of rectangular. Neither 
of these differences nor the offset of the mushroom follower affect 
the similarity of construction for the two types of cams. There- 
fore, only a brief review of the general method of construction for 
the present problem will be given here. 

112. Inasmuch as the line of pressure between cam and follower 
is always parallel to the direction of motion of the follower in prob- 
lems such as this, there is no pressure angle in the ordinary sense. 
If a computation for size of cam is made in the usual way, the radius 
of the pitch circle will figure to be unnecessarily large, due princi- 
pally to the fact that only a 45° degree turn of the cam is allowed 
for the upward motion of the follower. 

A radius A, Fig. 52, which allows for radius of shaft, thickness 
of hub, etc., is assumed, and the follower motion of 4 units is laid 
off at A V. This distance is divided into four unequal parts at 
H, I . . . which are to each other as 1, 3, 5, and 7, thus giving 
uniform acceleration all the way up. The angle A B oi the cam 
is laid off 45° and is divided into four equal time parts. The follower 
or toe surface yl IF is then moved up the distance A H and revolved 
through the angle A 1 to the position Hi H^ which is marked. Simi- 



62 



ELEMENTARY CAMS 



larly AW is next moved to I h and revolved to 7i h. The smooth- 
est possible convex curve is then drawn to the lines Hi H2, I1I2 . . . 
and this curve becomes the working surface of the wiper. 

The necessary working length for the wiper is found to be A V2, 
and, adding a small arbitrary distance, V2 C, the total length is taken 




Fig. 52. — Problem 12, Toe and Wiper Cam 

as A C. The total length of the toe A W will be equal to Vi C. 
The long dash lines in Fig. 52 indicate the highest position of the 
toe and wiper, and the short dash-line curve marks the locus of 
contact between the wiper and toe. This curve is obtained by 
making, for example, J J3 equal to Ji J2. 

113. Modifications of the toe and wiper cam. The toe and 
wiper cam constructions are commonly used. In the present ele- 
mentary problems the cam or wiper is assumed to oscillate with 
uniform angular velocity, whereas in practice it usually has a variable 
angular velocity due to the fact that it is operated through a rod 
which is connected at the driving end to a crank pin or eccentric 



CAM PROBLEMS AND EXERCISE PROBLEMS 63 

whose diameter of action corresponds to the swing of the wiper 
cam. The follower toe may be built with a curved instead of a 
straight line, by a sKght modification in detail which consists in draw- 
ing the curved toe line in place of the straight lines, H^ H2, Iih . . . 
as shown in Fig. 52. These points, together with a consideration of 
the amount of slip between the surfaces in this type of cam and a 
discussion of the necessary modification to secure pure rolling in 
cams of this general appearance, are subjects for more advanced 
work than is covered by the present elementary problems. 

114. Exercise problem 12a. Required a wiper cam to operate 
a flat toe follower which shall move: 

(a) Up 3 units with uniform velocity while the cam turns 60° 
in a counterclockwise direction with uniform angular velocity. 

(b) Down 3 units with uniform velocity while the cam turns 60° 
in a clockwise direction with uniform angular velocity. 

115. Yoke cams. Yoke cams are simple radial periphery cams 
in which two points of the follower, instead of one, are in contact 
with the working surface. The contact points are usually diametri- 
cally opposite to each other. Roller contact is generally used and 
the centers of the rollers are a fixed distance apart. The yoke cam 
gives positive motion in both directions, and does not depend on a 
spring or on gravity to return the follower as do all other cams 
thus far considered, excepting the face cam. 

116. Problem 13. Single-disk yoke cam. Required a single 
radial cam to operate a yoke follower with a maximum pressure 
angle of 30°: 

(a) Out 4 units in 45° turn of the cam, on crank curve. 

(b) In 4 " " 90° " " " " " " '' 

(c) Out 4 '' " 45° " " " '' '' " 

117. With a single radial cam for a yoke follower, data may be 
assigned only within the first 180°. The reason for this will appear 
presently. 

Compute the radius of pitch circle as in ordinary radial cam 
problems. It is found to be 13.86 units and is laid off at Z), Fig. 53. 
The pitch surface, A Di Fi Ai 72, is found in the usual way. Then 
the diametral distance, AF2, will be the fixed distance between the 
centers of the rollers, and if this distance is laid off on diametral lines, 
as from 7i, Kx . . ., the points TF, X ... on the complementary 
pitch surface will be located. A size of roller A 5 is next assumed 
and the working surface B Bi is constructed. The maximum radius 
of the working surface is finally located, as at B^. A small amount 



64 



ELEMENTARY CAMS 



is added to this for clearance and the total laid off at Z, thus giving 
the width of yoke necessary for an enclosed cam. 

118. Limited application of single-disk yoke cam. In yoke 
cams constructed from a single disk the data are limited in two ways : 

First, data can be assigned for the first 180° only, because the 
pitch surface for the second 180° must be complementary to the 
pitch surface in the first 180°. 

Secondly, the complementary pitch surface cannot approach any 
nearer to the center of rotation of the cam than does the pitch surface 



n Llii^ 



T 
I i 



,-f^r- 



pr 






'A', 

HI 




Fig. 53. — Problem 13, Single-Disk Yoke Cam 



in the first 180°, otherwise the follower will have a greater motion 
than that which was assigned to it. 

To illustrate this second case, assume that item (c) had been 
changed in the data for Problem 13 so as to specify that the follower 
should remain at rest while the cam turns 45°. Then the pitch 
surface of the cam for the first 180° would have been AViAiC, 
Fig. 54, instead of AViAi F2. The diametral distance A C would 
then have been the distance between roller centers, and would have 
been also the distance used in determining the complementary 
pitch surface C E1A3A which, it will be noted, approaches closer 
to than does AViC. When Ei of the complementary surface 



CAM PROBLEMS AND EXERCISE PROBLEMS 



65 




Fig. 54. 



reaches the center line D, the center A of the roller will be at E 
and the roller will have traveled the distance A E in addition to the 
travel A V which was assigned. Furthermore, the pressure angle 
will be very high when F crosses the line A. With the data 
which gives the pitch surface A Vi C, the yoke follower will move 
just twice the assigned distance. This double motion will not be 
continuous, as the follower 
will be at rest for a definite 
period represented by AiC. 
Even if the data were such 
that Ai should fall at C there 
would be a momentary pe- 
riod of rest for the follower 
at the middle of its stroke. 

Summing up, the desired 
travel, pressure angle, and 
follower velocity will be ob- 
tained in single-disk yoke 
cams, only when the data 
are such as to have the fol- 
lower at the extreme oppo- 
site ends of its stroke at the 

zero and 180° phases. In other cases increased travel, increased 
pressure angle, and irregular follower velocities will have to be 
considered. 

All of the limitations of the single-disk yoke cam may be avoided 
by using the double disk cam as illustrated in Problem 14. 

119. Exercise problem 13a. Required a single-disk radial cam 
to operate a yoke follower with a maximum pressure angle of 30°: 

(a) In 6 units in 60° turn of the cam on parabola curve. 

(b) Out 6 " " 45° " " " " " 

(c) At rest for 30° " " " " " 

(d) In 6 units in 45° " " " " " 

120. Problem 14. Double-disk yoke cam. Required a double- 
disk cam to operate a yoke follower with a maximum pressure angle 
of 30°: 

(a) To the right 6 units in 150° turn of the cam, on the crank 
curve. 

(b) To the left 6 units in 90° turn of the cam, on the crank curve. 

(c) To remain stationary for 120° turn of the cam, on the crank 
curve. 



-Illustrating IjImited Application of 
Single-Disk Yoke Cam 



66 



ELEMENTA.RY CAMS 



121. The detail of construction for the primary disk is the same 
as in previous problems involving radial cams. In this problem, then, 

the radius of the pitch circle is 6 X 2.72 X 4 X ttti X it = 10,4 

units and this is laid off at D, Fig. 55. The forward driving pitch 
surface, A HiVihA, is constructed in the regular way as indicated 
by the construction lines. 

122. The diameter D C oi the pitch circle is next taken as a 
constant and its length is laid off on diametral lines from successive 




Fig. 55. — Problem 14, Double-Disk Yoke Cams, Detail CoNSTRrcTiON 



points on the primary pitch surface, thus giving the secondary or 
return pitch surface. For example, the point P on the secondary 
surface is found by making A P = D C; the point M by making 
HiM = D C. . . . This second cam disk has a pressure angle of 
30° at Da, the same as the primary disk has at D2. Had any diam- 
etral length other than D C been taken in this problem as a constant 
for constructing the second cam, the pressure angle at D4, would 
have been greater or less than the assigned 30°. It does not follow 
that the diameter of the pitch circle should be used as a constant 
for generating the complementary cam. The determining factor, 
in selecting a constant diametral length is that the maximum pres- 
sure angle on the second cam should not exceed the assigned value. 



CAM PROBLEMS AND EXERCISE PROBLEMS 



67 



123. To avoid intricate line work, only the detail drawing for 
the construction of the pitch surfaces for this problem is shown 
in Fig. 55. The pitch surfaces are then redrawn in Fig. 56 and 
the working surfaces and the yoke constructed. 

The working surface of the primary or forward-driving cam is 
shown at B E F G B, Fig. 56, and is constructed in the same way as 




Fig. 56. — Problem 14, Double-Disk Yoke Cams Showing Strap Yoke and Rollers 



in previous problems by drawing it as an envelope to successive 
roller positions. The working surface of the return cam is shown 
at S Q R S. A special caution to be observed at this point is that 
the working surface of the second cam cannot be obtained directly 
from the working surface of the first cam by using the diametral 
constant; the second cam pitch surface must be obtained first. 

124. The form of yoke in yoke cams may vary, as illustrated for 
example by the box type which encloses the cam. Fig. 53, and by 
the strap type, Fig. 56. In the latter illustration the strap W X 
has a longitudinal slot T U permitting it to move back and forth 
astride the shaft without interference. The guide arms of the 



68 ELEMENTARY CAMS 

yoke are shown at Y and Z. In all yoke constructions it is desirable 
to have all the forces acting in as nearly a straight line, or in a plane, 
as possible. In Fig. 53 this is obtained, as may be noted in the 
top view where the longitudinal center lines of cam disk, cam roller, 
yoke and yoke guides are all in the same plane. In Fig. 56 the yoke 
guides, F' and Z', are placed in a line lying between the cam disks, 
jB' and S\ so as to have the forces balanced to a greater degree than 
they would be if the guides were in line with the strap W^ X' . 

125. Exercise problem 14a. Required a double-disk cam to 
operate a yoke follower with a maximum pressure angle of 30°, as 
follows : 

(a) To the right 4 units in 90° on the parabola base. 

(b) Dwell for 30°. 

(c) To the right 4 '' " 105° '' " 

(d) " " left 8 " " 135° " " 

126 Problem 15. Cylindrical cam with follower that 
MOVES in a straight LINE. Required a cylindrical cam to operate 
a reciprocating follower rod : 

(a) To the right 4 units in 120° on the crank curve. 

(b) " '' left 2 " " 120° " '' " 

(c) " dwell 120°. 

The maximum surface pressure angle to be 30°. 

127. The size of cylinder is found by a computation similar to 
that for radial cams, and in this problem the radius of the cylinder is, 

4 X 2.72 X 3 X 3-j^ X 2- = 5.2 units. 

This distance is laid off at O' A' in Fig. 57, and the circle drawn. 
The distance A V, the travel of the follower, is laid off equal to 
4 units and subdivided, according to the crank circle, 2X H, I . . . 
The radius of the follower pin is assumed as at A ;S and this distance 
is laid off at C S, thus locating the edge of the cylinder. Make 
V D equal to A C. The circle representing the cylinder is next 
divided into three 120° divisions Bit A', M\ and Q', as specified. 
A' M' is divided into six equal parts by the points H' , I' . . . 
which are projected over to meet the vertical construction lines 
through H, I ... Sit H2, h- - - • The latter points mark a curve on 
the surface of the cylinder. This curve is a guide for the center of 
the tool which cuts the groove. The finding of this curve and the 
construction of the follower pin and rod constitute the remaining 
essential work on this problem. If it is desired to show the groove 



CAM PROBLEMS AND EXERCISE PROBLEMS 



69 



itself, the directions in paragraph 134 will give an approximate 
method. The follower pin is attached to a follower rod X which 
is guided by the bearings Y and Z. The assigned pressure angle 
of 30° is shown in its true size at D J G; J D being parallel to the 
direction of motion of the follower rod, and D G being a normal 
to the cutting-tool curve M N J P. . . . In general, the pressure 
angle will not show in its true size, and if it is then desired to illus- 
trate it, the cylinder may, in effect, be revolved until the correct 
point of the cutting-tool curve is projected on the horizontal center 
line. The exact point E where the cutting-tool curve comes tangent 
to the bottom line of the cylinder may be found by locating Ei 
relatively to Ki and Li, the same as E' is located relatively to K^ 
and L', and projecting Ei down to E. 

A small clearance is allowed between the end B' of the pin and the 
inner surface of the groove, which is represented by the dash circle 
passing through F\ 

128. Refinements in cylindrical cam design. It will be 
noted that the '^ maximum surface pressure angle" was given in the 
data for this problem instead of the term ''maximum pressure 
angle" that has been used thus far. The reason for this is that 
the pressure angle varies along the length of the pin and is always 
greatest at the inner end, that is, at the point B in Fig. 57. This is 
not important in most practical cases. Further, the term ''pitch 
cylinder" is not mentioned in the simple form of practical construc- 




FiG. 57. — Problem 15, Cylindrical Cam with Follower Sliding in a Straight Line 



70 ELEMENTARY CAMS 

tion here used. Since the pitch cyUnder should pass through the 
point where maximum pressure angle exists, the pitch cylinder 
in cams of this type would be one having a radius O' B' . The pitch 
surface of the cylindrical cam would be a warped surface, known 
as the right helicoid, and the intersection of this surface with the 
surface of the cylinder is the curve R A E P R and is the guide 
curve for the cutting tool in milling out the groove for the pin. 
The sides of the groove are the working surfaces of the cam; they 
are indicated in the sectioned part of the front view of Fig. 57. 

More exact methods for drawing the sides of the groove in a 
cylindrical cam, together with a more exact method for determining 
the maximum pressure angle, involve a knowledge of projections 
and an intricacy in drawing that make such work a proper subject 
for advanced study, and it will therefore be omitted in this elemen- 
tary treatment, as it is totally unnecessary in most practical work. 

129. Exercise problem 15a. Required a cylindrical cam to 
operate a sliding follower rod, with a maximum pressure angle of 40° : 

(a) 6 units to the right in 90° on the parabola base. 

(b) 6 " " " left " 270° " " 

130. Problem 16. Cylindrical cam with swinging follower. 
Required a cylindrical cam to operate a swinging follower arm: 

(a) To the left 40° in 120° turn of the cam on the crank curve. 

(b) '' " right 40° " 120° " " '' " " " 

(c) Dwell for 120° turn of the cam. 

The length of the follower arm is to be 9 units and the approximate 
maximum pressure angle is to be 30°. 

131. The diameter for the cylindrical surface is found in the 
same manner as the diameter of the pitch circle in radial periphery 
cams. The data in this problem do not give directly the travel of 

the follower and so this value must be 
found first. The chord of a 40° arc hav- 
ing 9 units radius will be 

9 X 2 X sin 20° = 9 X 2 X .342 = 6.2 units. 

^ If a ' trigonometrical table is not at 

Fig. 58. — Determining Fol- i i ,i it , • 

LOWER Travel IN Swinging hand the arc may be drawn out as m 
dSmfX.'^''''''''^'''''^'"''''' Fig- ^8 where half the given angle is laid 
out at AY J by the simple expedient of 
subdividing a 30° arc by means of a dividers. The half chord 
A D is drawn and measured. It is equal to 3.1 units, thus mak- 
ing the chord of the whole arc of travel equal to 6.2 units. 




CAM PROBLEMS AND EXERCISE PROBLEMS 



71 



This value is used in obtaining the diameter of the surface of the cyl- 
inder as follows: 

1 



6.2 X 2.72 X 3 X 



3.14 



16.12 units. 



The circle A' Q' M' , Fig. 59, is drawn with a radius of 8.06 units. 

132. The 120° angles assigned in the data are next laid out but 
not from the center line R' as in previous problems. In mechan- 
isms of all kinds where there is a swinging follower, it is a rule, unless 
otherwise specified, that the swinging pin should be the same dis- 




FiG. 59. — Problem 16, Cylindrical Cam with Swinging Follower Arm 

tance above a center line at the middle of its swing as it is below 
at the two extremities of its swing. In this case, then, the point G, 
Fig. 58, will be marked midway between J and D and the distance 
G J laid off at G J in Fig. 59. J will be the center of swing of the 
follower arm and the arc of swing of the follower pin will h^ A J V. 
J will be as much above the center line as A and V are below. The 
practical advantage of this detail in the layout is that it gives a 
maximum bearing length between the follower pin and the side of 
the groove. 

133. The arc A J V, Fig. 59, is next divided at the points marked 
H, I . . . according to the crank curve assignment, and vertical 
construction lines are drawn through these points. 

The point A is now projected to A' and the radial line, A' 0, is 
drawn. This becomes the base line from which to lay off the three 



72 ELEMENTARY CAMS 

assigned timing angles of 120°, as shown at A' M', M' Q\ and 
Q' A\ The arc A' M' is next divided into the desired number 
of equal construction parts, as at Hs, h, Js. . . . 

When Hz reaches A', the pin A will have swung not only over 
to -H", but it will have moved up the distance A' H' measured on the 
surface of the cylinder. Therefore, when Hz reaches A' , it is the 
line through i^s (^3 H^ = A' H') on the groove center line that will 
be in contact with the pin center line. For this reason H^, instead 
of Hz, is projected over to meet the construction line at H2. This 
latter point is on the guide curve for the cutting tool on the surface 
of the cylinder. Other points are found in the same way. Time 
may be saved by marking the points A' H' I' J' on the straight edge 
of a piece of paper and transferring these marks at one time so as 
to obtain the points Is, J5 . . • Pb- - • . 

134. If it is required to show the surface bounding lines of the 
side of the groove it may be done quickly, although approximately, 
by laying off on a horizontal line, as at I2, the points h and h at 
distances equal to the radius of the pin. These will represent points 
on the curve. If it is required to show the bottom lines of the 
groove it may be done by projecting from 1 7 and finding, for 
example, the point h in the same way as h was found. 

135. Exercise problem 16a. Required a cylindrical cam to 
operate a swinging follower arm: 

(a) To the right 6 units (measured on chord of follower pin arc) 
while cam turns 150°. 

(b) Dwell while cam turns 120°. 

(c) To the left 6 units while cam turns 90°. 

The follower arm to be 8 units long and its rate of swinging to be 
controlled by the crank curve with a maximum approximate pres- 
sure angle of 40°. 

136. Chart method for laying out a cylindrical cam with 
A swinging follower arm. This method is illustrated in Figs. 60 
and 61. The data in this problem will be taken the same as in 
Problem 16, namely, that a follower arm of 9 units length shall: 
Swing through an angle of 40° to the left while the cam turns 120°; 
through the same angle to the right while the cam turns 120°, on 
the crank curve in both directions; remain stationary while the 
cam turns 120°. The maximum pressure angle is to be approxi- 
mately 30°. 

137. To find the length of the chart, the chord that measures 
the arc of swing of the follower pin is first determined to be 6.2 



CAM PROBLEMS AND EXERCISE PROBLEMS 



73 



Hi 



-y. 



Mi 



Y, 



A. 



«4 



units as explained in paragraph 131. The length 
of chart is 

6.2 X 2.72 X 3 = 50.6 units, 

and this is laid off at J Ji, Fig. 60. The length 
of the follower arm is then laid off at J Y, and 
the follower-pin arc A V drawn. This arc is 
subdivided at H, 7 . . . according to the crank 
curve. The distance Y Fe is then laid off to 
represent 120° and its length will be equal to 
one-third the length of the chart. As many 
construction points as were used from A to V 
are then laid off between Y and Fe. With 
these as centers and YA as a radius draw a series 
of arcs to which the points H, I . . . are pro- 
jected, thus giving the base curve through the 
points Hi, 1 1. . . . Tangent to the series of 
arcs on the chart draw straight lines and mark 
the intercepts H4H2, hh. . • . 

138. Upon completing the chart, the surface 
of the cam is drawn as in Fig. 61, with a diam- 



eter E' T' = 



50.6 
3.14 



16.12. The width C iV of 



the cylinder may be taken equal to the chord 
A F of the arc of swing of the follower pin, plus 
twice the diameter of the pin. 

139. The simplest general plan for trans- 
ferring the cam chart to the surface of the 
cam is to consider the chart lines to be on a 
strip of paper, and that this paper is simply 
wound around the cylindrical surface of the 
cam, starting the point G of the chart at G on 
the center line of the cam. G on the chart is 
midway between J and D, Then the points 
H2, h ' ' ' oi the base curve in Fig. 60 will fall 
at H2, 1 2, in Fig. 61, giving the surface guide 
curve for the center of the cutting tool. 

140. The detail necessary to actually lo- 
cate the points H2, h in Fig. 61 is accomplished 
by projecting J to J' and laying off the as- 



74 



ELEMENTARY CAMS 



signed 120° divisions, and also the subdivisions from this latter 
point. The 120° divisions are shown at M\ Q', J'; the equal 
subdivisions at H3 13. . . . From these latter points, lines are pro- 
jected to the front view and the lengths i^4 H2, li I2 are transferred 




Fig. 61. — Cylindrical Cam with Swinging Follower Drawn from Chart 



from Fig. 60. To find the point of tangency at E, make K4. Ei 
of Fig. 60 equal to Kz E' of Fig. 61, then draw Ei E in Fig. 60 
and lay off this distance from the center line G F in Fig. 61, thus 
giving the point E. To find the point of tangency at M, lay off 
at M' Ms a distance equal to the chart distance from M2 to M 
in Fig. 60 and project M3 of Fig. 61 to M. 



SECTION IV.— TIMING AND INTERFERENCE OF CAMS 



141. In machines where two or more cams are employed it is 
generally necessary to lay down a preliminary diagram showing 
the relative times of starting and stopping of the several cams, in 
order to be assured that the various operations will take place in 
proper sequence and at proper intervals. The same preliminary 
diagram is also used to avoid interference and to make clearance 
allowances for follower rods whose paths cross each other. 

142. Problem 17. Cam timing and interference. Required 
two cams that will operate the follower rods A and E, Fig. 62, lying 
in the same plane, so that: 

(a) Rod A shall move 16 units to D, dwell for 30°, return 8 units 
to B and again dwell 30°, all to 
take place in 180° turn of the 
cam. The cam to produce the 
same motions in the second 180° 
but in reverse order. 

(b) Rod E shall cross path 
of rod A and move 4 units be- 
yond it and back again during 
the time that rod A is moving 
Irom D to Bio D. 

All motions to be on the 
crank curve with maximum pres- 
sure angles of 40°. 

143. Before taking up the 
solution of this problem in de- 
tail it should be noted: 1st, that 
any convenient type of cam may 
be used in problems of this kind ; 

2d, that usually only general motions of followers or objects are given 
in the preliminary data, as above, and that the cam designer must 
supply data and restate the problem in terms of angles for each of 
the movements after studying the preliminary data with the aid of 
a timing diagram. 

144. The first step leading to a restatement of the problem is to 
determine the number of degrees in which rod A may move the 

75 




Fig. 62.— Problem 17, Preijminaky Layout 
OF Data for Problem in Cam Inter- 
ference 



76 



ELEMENTARY CAMS 



16 units, and also the number of degrees in which it may move 

the 8 units in order that the pressure angle will be 40° in both cases. 

Since there are two 30° dwells in the first 180° there will be 120° 

left for the two motions of which the first 

require ht of 120"^ 




motion will 
second, 40°. 



24 



or 80°, and the 



The length of chart for 



cam A 

may now be computed as 16 X 1.87 X -^tt 

= 134.6 and laid off as at A Ai, Fig. 63. The 
height of the chart AD is 16 units. The 
chart is next divided into degrees of any con- 
venient unit, 0, 10°, 20° . . . being used in 
this case. For the present the base line may 
be made up of a series of straight lines as at A 
A, Di D2, D2 5i. . . . 

145. The amount of clearance between 
the moving arms must now be decided upon. 
Let it be the designer's judgment that the end 
of the follower rod E should lie at rest 1 unit 
to the left of rod A as shown in Fig. 64, and 
that rod E should not begin to move until 
the rod A is one unit out of the way. Then 
A will be at C, Fig. 64, moving down, when 
E starts, assuming the rod E to he 3 units 
wide and that it is so placed that its top edge 
is one unit below D. The point C is then 5 
units from the top of the stroke and if this 
distance is laid off in Fig. 63, as shown, the 
fine C Ci is obtained cutting the crank curve, 
which should now be drawn at C. C is at 
the 133° point and this, then, is the time when 
the follower E should start to move. 

146. The total motion for rod E is 4 + 5 + 
1 = 10 units, assuming width of rod A to be 5 
units. The time during which this motion can 
take place, outward, is 180° - 133° = 47° as 

represented at Ei E2, Fig. 63. If the crank curve Ei F is now drawn it 
will be intersected by the one-unit clearance line GiG a,tG which rep- 
resents, in this case, a rotation of approximately 11° of the cam that 
drives rod E. The total clearance for the two rods which cross each 



Fig. 63. — Problem 17, 
Timing Diagram for 
Avoiding Interfer- 
ence OF Cams 



TIMING AND INTERFERENCE OF CAMS 



77 



other's paths is now found to be 3° for cam follower A and 11° for cam 
follower E, or 14° of the machine cycle. These clearances are in- 
dicated in Fig. 63. If it is the judgment of the designer that errors 
in cutting keyways and in assembling, and that the wear of the parts 
will fall within these limits, the cams may now be drawn. 

147. The cam chart for cam E was made the same length as 




Fig. 6-1. — Problem 17, Design for Definite Timing and Non-interference of 
Cams Operating in Same Plane 



the chart for cam A in order to make clearance allowance. The 
true length of this chart, for a 40° pressure angle, would be: 

10 X 1.87 X ^ = 143.2 units, 

instead of 134.6 as now drawn. If an exact clearance allowance 

in degrees were required, it would be necessary to redraw the crank 

47 
curve El F, making the distance E1E2 equal to w^ of 143.2. It is now 

47 

oTTTj of 134.6. With a new and exact drawing the crank curve Ei G 



78 



ELEMENTARY CAMS 



Avould not rise quite so rapidly and the intercept at G would show a 
small fraction over the 11° taken above. In some problems where the 
lengths of the true charts differ considerably it may be necessary to 
redraw this part of the base curve to be sufficiently accurate in 
obtaining the clearance in degrees. 

148. The radius of the pitch circle for the cam operating rod A 
134.6 



will be 



6.28 



21.4 units as drawn at H 7, Fig. 64. The pitch 



-7^ = 22.8 and this is laid off at M S. The location of M and the 



1 1 A 

1 1 

\ D \ 

! 1 




' 






1 ^ j 1 « 1 ■ 
L J L i 


L 



surface of the cam and the working surface are drawn in the same 
way as the ordinary radial cams in previous problems. The length 
of the rod Ai A may be assumed. 

The radius for the pitch circle for the cam operating rod E will be 
143^ 
6.28 

length of the rod N E will either enter into the layout of the frame- 
work of the machine in a practical problem, or will be determined 

by the framework if pre- 
viously laid out. In the 
present case it will only 
be necessary, in deter- 
mining the length of rod 
N E and the position 
of M, to make certain 
that the shafts M and H 
are sufficiently far apart 
to keep the cams from 
striking when turning. 

149. Location of keyways. It is important to locate the 
keyway exactly by giving its position in degrees so as not to destroy 
the clearance values already made. Since the working surfaces of 
cams frequently approach close to the hub or shaft it is a good 
plan to place the keyway at the center of the longest lobe of the 
cam, as illustrated in both cams. 

150. Exercise problem 17a. Assume a stack of blocks at A, 
Fig. 65. Required that the bottom block shall be delivered with 
one stroke at C, the next block at £>, being moved first to B and 
then to D, the next block at C, the next at D, etc. Let the sizes of 
the blocks and the distances they must be moved be as shown in 
Fig. 65. Lay out cam mechanism to secure this result, keeping 
the maximum pressure angle at 30°. 



Fig. 65. 



-Problem 17a, Diagram Showing Appli- 
cation OF Data 



SECTION v.— CAMS FOR REPRODUCING GIVEN 
CURVES OR FIGURES 

151. Problem 18. Cam mechanism for drawing an ellipse. 
Required a cam mechanism that will reproduce the ellipse AC B D 
in Fig. 66, the marking point to move slowly at the extremities A 
and B of the major axis and rapidly at C and D, the rate of increase 
and decrease of velocity being uniform. 

152. Divide A C into three parts which are to each other as 1, 
3, and 5; C B into three parts which are as 5, 3, and 1 ... in order 
that the marking point shall move through increasing spaces in 
equal times in moving from A to C. . . . For greater accuracy A C 
would be divided into a greater number of parts. 

153. In devising the mechanism assume that the marking point 
shall be at the end of a rod which shall be controlled by two com- 
ponent motions that are horizontal and vertical, or nearly so. This 
suggests the rod A E F, with marking point at A, with horizontal 
motion supplied from a bent rocker attached at F and with vertical 
motion supplied from a reversing straight arm rocker L K J, attached 
through a link E J Sbt the point E. The lengths of the links and of 
the arms of the rockers, and the positions of the fixed centers of the 
rockers will have to be assumed, the lengths of the arms and links 
being such that none of them will have to swing through more than 
60°. With more than 60° swing the angle between an arm and a link 
is liable to become too acute for smooth running. Where rocker 
arms are connected to links the ends of the rocker should, in general, 
swing equal distances above and below the center line of the link's 
motion, as for example, the points F and 6 on the arc of swing of F 
should be as much above the line ^ M as the point 3 is below. Also 
the arc 3 J 9 should swing equally on each side of J T in order to 
secure best average pressure angles for the mechanism. 

154. Let each of the rocker arms be assumed to be controlled by 
single-acting radial cams. The center of roller H will be required 
to swing on an arc 6 H which, continued, passes through M. This 
gives small pressure angles while A is traveling to B, especially when 
^ is at C and is moving fastest. It gives large pressure angle, how- 
ever, while A is traveling from B to D to A. If A is assumed to do 
heavy work along A C B and to run light along B D A this is the 

79 



80 ELEMENTARY CAMS 

better arrangement. If A did the same work on both strokes it 
would be better to place the rocker arm G H so that H and 6 rested 
on a radial line. The center of roller L will be assumed to travel 
on an arc whose extremities are on a radial line, or nearly so. 

With A F as Si radius and A, 1, 2 ... as centers, strike short 
arcs intersecting F 6 at F, 1, 2 . . . numbering the arcs as soon as 
drawn to avoid confusion later on. Lay off points on H 6 corre- 
sponding to those on F 6. 

155. Inasmuch as the point H does not move in accordance with 
the law of any of the base curves no precise computation can be 
made for the size of the pitch circle for any given pressure angle 
and it may be omitted. Instead, a minimum radius M H oi the 
pitch surface may be assumed. If it is desired to control the pres- 
sure angles it may be done by first constructing the pitch surface, 
H V W, and then measuring the angles at the construction points. 
Some of these are shown in Fig. 66, at H, 3, 6, and 8, and are 20°, 
— 12°, 48°, and 57°, respectively. If these angles should prove un- 
satisfactory a larger pitch circle, or a differently proportioned rocker, 
may be used. Or, an approximate computation for radius of pitch 
circle by the method which is explained to advantage in connection 
with the next problem, paragraphs 164 and 165, may be used. 

156. To construct the second cam, take the distance A E as a 
radius and A, 1, 2 . . . as centers and mark the points E, 1,2. . . . 
Again, with the latter points as centers and E J as a radius, mark 
the points J, 1, 2 . . . and transfer these to L, 1, 2. . . . With 
the latter points marked, the pitch surface of the second cam, P QR, 
is constructed in the same way as was the first cam. 

The angle between the keyways, marked 393^2° iii Fig. 66, must 
be carefully measured and shown on the drawing. 

157. Exercise problem 18a. Required a cam mechanism that 
will draw the numeral 8, the marking point moving with uniform 
velocity. 

158. Problem 19. Cams for reproduction of handwriting. 
Required a cam mechanism to reproduce the script letters S t e. 

159. The first step in the solution of this problem is to write the 
letters carefully, for if the machine is properly designed it will re- 
produce the copy exactly as written. The copy is written at A in 
Fig. 67. 

160. The next step is to decide on the kind of mechanism and 
the type of cams to be used, for the problem may be solved by a 
number of different combinations. The mechanism for this problem 



-i 


F5l 


/ 


( 




N 


I 


^ 


? 
y 

«> 



^ Ci 


\ 


T 


^. 


1 


J-^ ©» 



^N<00y 




/ ^ 


r 


'' 


« 




Fig. 66. — Problem 18, Cam for Drawing an Ellipse 




■h -f 
PiG. 67. — Problem 19, Cams for Reproducing Script Letters, etc. 



CAMS FOR REPRODUCING GIVEN CURVES OR FIGURES 8^ 



i 




will consist of two radial single-acting cams 
mounted on one shaft, and a swinging rocker 
arm mounted on a pivot which is moved forth 
and back on a radial line as shown in Fig. 67. 
This mechanical combination is selected for this 
cF|l> problem because it involves methods of construc- 

tion not used in any of the preceding problems. 

161. The actual work of construction is started 
by marking off a series of dots along the lines of 
the entire copy, as shown at A, and marked from 
zero to 64. Inasmuch as there is some latitude 
in the spacing, and consequently in the nmnber 
of these dots, as will be explained presently, it is 
advisable to use a total number of dots whose 
least factors are 2 and 2, 2 and 3, or 2 and 5. 
This is not essential but it will faciHtate the work 
later on. 

162. The matter of placing the dots is per- 
haps the most important item of the entire prob- 
lem, for on this depends the size of the roller and 
smooth action. In fact, with some methods of spac- 
ing, no roller can be used at all and a sharp V- 
edge sliding follower will have to be used if true 
reproduction is desired. 

The basic considerations in selecting the 
points are: 

First, that a point should be located at the 
extreme right and extreme left of each right and 
left throw, as at - 7, 7 - 16, 16 -20 ... in Fig. 
67, A, and at the top and bottom of each swing, 
BLS Sit - 8,8 - 13, 13 - 18 . . .; and. 

Secondly, that the marking point should start 
slowly and come to rest gradually on each stroke, 
considering both of the component directions of 
its motion at the same time. On account of this 
it is impossible to secure ideal conditions at all times and com- 
promises must frequently be made. For example, the component 
motions of the marking point D are: First, a horizontal one due 
to Cam No. 1 ; and secondly, a vertical curvilinear one due to Cam 
No. 2 and the rocker arm H G D. The intermediate points 0-7 on the 
upper swing of the letter S are so selected as to give increasing and 



84 ELEMENTARY CAMS 

decreasing spaces in the horizontal projections on D E, and the 
same points, together with point 8, are selected at the same time so 
as to give increasing and decreasing spaces when projected onto the 
arc D F. Each space between a pair of adjacent numbers represents 
the same time unit. On this basis the entire spacing of the copy 
is done. 

163. With each of the points in the group at A, Fig. 67, as centers, 
and with a radius, D G, mark very carefully the corresponding points 
on G L in group B. To avoid confusion it is essential here to adopt 
some method of identifying points so marked for later reference. 
A satisfactory method is shown at B, all the motions to the right 
being indicated below, and the motions to the left, above G L. 

164. The sizes of the cams are to be next computed. To do 
this select the largest horizontal space in section A. This is found 
between 56 and 57 and is equal to .46 of the unit of length that 
happened to be selected in this problem. Assuming that the marking 
point moves with uniform velocity over this distance, and that a 
pressure angle of 40° is suitable in this instance where no heavy 
work is done, the factor of 1.19 is taken from the table in paragraph 
30. Since there are 64 time units the length of circumference of 
pitch circle for Cam No. 1 will be 

.46 X 1.19 X 64 = 35.03, and the radius 5.58. 

165. Before calculating the size of Cam No. 2 the length of the 
rocker arm G H must be decided upon and this will be taken in this 
problem at 5 units, the same as the arm G D. Then the total swing 
of the follower point H will be H K, equal to D F, and the greatest 
swing in any one direction in any one time unit will be during the 
periods 10-11 and 61-62, shown at A, Fig. 67, both equal to .48 units. 
Making the same computation as for Cam No. 1, 

.48 X 1.19 X 64 



3.14 X 2 



= 5.82 



equals the pitch radius of Cam No. 2. 

166. The position of the cam shaft relatively to the pivot 
arm G depends on what is desired for the position of the arc H K 
with reference to the cam center. If it is desired that the points 
H and K shall be on a radial line from the center of the cam, which 
gives best practical average results for both in and out strokes, 
proceed as follows : Draw chord D F at A in Fig. 67 ; bisect it at 
J and measure distance G J which is 4.93 units. Then the distance 



CAMS FOR REPRODUCING GIVEN CURVES OR FIGURES 85 

G will be the hypothenuse of a right angle triangle of which one 
side is 4.93 and the other 5.82. This may be separately drawn and 
the length of the hypothenuse found graphically or it may be 
figured as follows : 



GO = V 5-822 + 4.932 = 7.63. 

167. The pitch circles for both cams may be taken in problems 
of this kind to pass through the midpoint of the total travel. Then 
O M \& the radius of the pitch circle of Cam No. 1 and N P the 
total range of travel of the roller center; and Ji is the radius of 
the pitch circle of Cam No. 2 and H K the total range of travel of 
the roller center relatively to G. 

168. To find points on the pitch surface of the cams proceed in 
the usual way for Cam No. 1, by dividing the circle whose radius is 
N into as many equal parts as there were dots on construction 
points at A. Draw radial lines, and on these lay off the distances 
secured from B in Fig. 67 ; for example, the distance 3 Ni is laid 
out equal to G 3. The point Ni] and other points secured in similar 
manner, will lie on the pitch surface of Cam No. 1. 

169. The construction of the pitch surface for Cam No. 2 is 
different from that of Cam No. 1, and is different also from anything 
done in the preceding problems. In this case the resultant motion 
of the arm G H is made up of rectilinear translation and rotation and 
both components must be considered in laying out the pitch surface, 
for example, as follows : With G H as sl radius and point 4 of B as 
a center draw an arc intersecting the horizontal line through H at 4- 
Then when G is moved to 4. by Cam No. 1, H would be at 4 if the 
rectilinear component motion due to cam No. 1 were the only one 
acting. During the period represented by G 4, however. Cam No. 2 
must move the rocker arm through an arc Q 4, shown at A, and this 
arc must now be laid off at 4 R- The point R is then revolved to 
its proper position at T as follows : Divide the circle G into 
sixty-four equal parts. This is readily done in this problem because 
G is taken on the same radial line with N and the radial divisions 
already made on the circle having AT for a radius need only be 
extended. Lay off the distance G 4 ^^ 4 S- With aS as a center and 
G // as a radius draw the arc 4 T. Then T will be a point on the 
pitch surface of Cam No. 2. 

170. Having determined the pitch surfaces of the two cams the 
largest possible roller for each is found by searching for the shortest 
radius of curvature on the working side of each pitch surface. For 



86 



ELEMENTARY CAMS 



Cam No. 1 the size of the largest roller that can be used is that of 
the circle whose center is at U; and for Cam No. 2 it is that of the 
circle whose center is at V. In order to avoid sharp edges on the 
cams, rollers slightly smaller than these circles will be used. 

171. For assembling the cams the angles between them and the 
angles for the keyways should be carefully measured and placed on 
the drawing as shown in Fig. 67. 

A front view showing the elevations of the cams, lever arm, slide, 
and plate is given in Fig. 68. 

172. Method of subdividing circles into any desired num- 
ber OF EQUAL PARTS. The matter of subdividing the circle having 



C M 




Fig. 69. — Method of Subdividing Circles into Any Desired Number of Equal Arcs 



radius N, Fig. 67, into sixty-four equal parts was a simple matter 
of subdivisions. If it is required to divide the circle into eighty-seven 
equal parts the work is just as simple if a proper start is made as 
follows : Let it be required that the circle B D, Fig. 69, be divided 
into eighty-seven equal parts. Find the number next lower than 
eighty-seven whose least factors are 2 X 2, 2 X 3, or 2 X 5. Such a 
number is 80. Assume that the circle is 6 inches in diameter; then 
the circumference is 18.84 inches and /{y of this is 1.516 inches, 
which is laid off to scale on the tangent at B F. With a pair of small 
dividers, set to any convenient small measuring unit, step off divisions 



CAMS FOR REPRODUCING GIVEN CURVES OR FIGURES 87 

from F to the next step beyond B. Assume that there are 11 steps 
from F to G, then go forward 11 steps on the arc to K. Divide the 
large part of the circle K D B into eighty parts by the process of sub- 
division with the dividers as indicated by the divided angles 80, 40, 
20, 4, 2, and 1, in Fig. 69. Then B H i^ )^o of KDB, or )i^ of 
the entire circle, and the length B H will go exactly seven times into 
the arc B K. In this work nothing is said of the use of a protractor 
for laying off a large number of small subdivisions on a circle, al- 
though it may be used. The process of subdivision, however, always 
using the small dividers, gives automatically remarkably accurate 
results. 



INDEX 



A PAGE 

Acceleration given by different 
base curves 20-24 

Adjustable cams 11 

Angle in pitch surface of cam, 
effect of 37 

Automatic work, cams for 75 

B 

Barrel cam 7 

Base curve 14 

Base curves charted 15 

Base curves, method of construction 20 

Base line 14 

Box cam 8 

C 

Cam chart 12, 29 

Cam chart diagram 12 

Cam chart, omission of 31 

Cam considered as bent chart 34 

Cam defined 1 

Cam factors 17 

Carrier cam 11 

Chart showing cam factors 18 

Clamp cam 11 

Classification of cams 1, 12 

"Combination curve," straight 

line 15, 20, 56 

Conical cams 2, 11 

Crank curve 15, 21 

Cylindrical cam design, refinements 

in 69 

CyHndrical cam, straight moving 

follower 68 

Cylindrical cam, swinging follower 70 

D 

Diagram, cam chart 12 

Diagram, timing 13, 14, 76 

Disk cams 1 



PAGE 

Dog cam 11 

Double acting cams 9 

Double disk yoke cam 65 

Double end cam 7 

Double step cam 39 

Drum cam 7 

E 

ElHptical base curve 15, 23 

Empirical design 25 

End cam 7 

Exercise problems la to 18a 31-80 

F 

Face cams 3, 55 

Factors for pressure angles 17, 18 

Flat surface followers, Umited use 

of 49, 59 

Flat surface reciprocating follower 45 
Follower roller, limiting size of 35 
Frog cam 3 

G 
Groove cam 3 

H 

Heart cams 3 

Handwriting, cams for reproducing 80 



Interference of cams 75 

Internal cam 8 

L 

Limited application of single disk 

yoke cam 64 

Limited size of roller 35 

Limited use of flat surface fol- 
lower 49, 59 



89 



90 



INDEX 



PAGE 

Location of chart pitch line for 

double-step cams 39 

Location of swinging follower arms 53 

M 

Maximum pressure angle 17 

Maximum pressure angles for mul- 
tiple-step cams 41 

Multiple-mounted cams 11 

Mushroom cams 3, 45 

N 

Names of base curves 14 

Names of cams 2 

Non-circular arcs, radius of cur- 
vature of 38 

O 

Offset cam S 

Offset flat face follower 45 

Offset follower roller 42 

Omission of cam chart 31 

Oscillating cams 11 

P 

Parabola base curve 15, 22 

Periphery cam 2 

Pitch circle 15 

Pitch hne 15 

Pitch line for double-step cam .... 39 

Pitch point 16 

Pitch surface 15 

Plate cams 3 

Plate groove cams 3 

Positive drive cam 8, 55 

Pressure angle 17 

change in passing from chart 

to cam 33 

depends on size of pitch circle 33 

for flat face follower 48 

for offset foUower roller 43 

for swinging f oUower arms . . 50 

in multiple-step cams 41 

unequal on the two strokes. . . 31 
Problems 1 to 18 25-79 



R PAGE 

Radial cams 1 

Radius of curvature of non-circular 

arcs 38 

Reproduction of curves and figures 

by cams 79 

RoUer, Hmiting size of 35 

Rollers, for positive-drive cams. ... 37 

Rolling cams 5 

S 

Side cams 1 

Single-acting cams 9 

Size of roUer for cams 35 

Step cams 9 

Straight-hne base 15, 20 

Straight-hne combination curve 

15, 20, 56 

Strap cams 11 

Subdividing circles and arcs 86 

Swinging foUower arms, effect of 

flat surface contact 57 

location of 53 

roller contact 50-56 

T 

Technical design 27 

Time charts 13, 14, 76 

Timing of cams 75 

Toe and wiper cams 7, 61 

V 

Velocities given by different base 
curves 20-24 

W 

Wiper cam 5, 61 

Working surface 16 

Y 

Yoke cam 5 

double-disk 65 

single-disk 63 



'Mi 

til 



i 



J AHi WVtL'i t 






021 213 118 



m 



UktSi'Miii&'iii^i 



